diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index f00fd06..db55113 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-03T19:33:32","documenter_version":"1.5.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-07T06:39:13","documenter_version":"1.5.0"}} \ No newline at end of file diff --git a/dev/assets/spinspace/addition02.png b/dev/assets/spinspace/addition02.png new file mode 100644 index 0000000..d330e39 Binary files /dev/null and b/dev/assets/spinspace/addition02.png differ diff --git a/dev/assets/spinspace/addition08.png b/dev/assets/spinspace/addition08.png new file mode 100644 index 0000000..a6cb20c Binary files /dev/null and b/dev/assets/spinspace/addition08.png differ diff --git a/dev/assets/spinspace/addition09.png b/dev/assets/spinspace/addition09.png new file mode 100644 index 0000000..d0d43b3 Binary files /dev/null and b/dev/assets/spinspace/addition09.png differ diff --git a/dev/hopffibration.html b/dev/hopffibration.html index 01ba1b6..4319de7 100644 --- a/dev/hopffibration.html +++ b/dev/hopffibration.html @@ -172,4 +172,4 @@ updatecamera() end

To create an animation you need to use the record function. In summary, we instantiated a Scene inside a Figure. Next, we created and animated observables in the scene, on a frame by frame basis. Now, we record the scene by passing the figure fig, the file path of the resulting video, and the range of frame numbers to the record function. The frame is incremented by record and the frame number is passed to the function write to animate the observables. Once the frame number reaches the total number of animation frames, recording is finished and a video file is saved on the hard drive at the file path: gallery/planethopf.mp4.

    GLMakie.record(fig, joinpath("gallery", "$modelname.mp4"), 1:frames_number) do frame
         animate(frame)
-    end
+ end diff --git a/dev/index.html b/dev/index.html index 0dc1d04..2a397fd 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · Porta.jl
GitHub

Geometrize the quantum!

This project is inspired by Eric Weinstein's Graph-Wall-Tome (GWT) project. Watch visual models on the YouTube channel.

Requirements

  • CSV v0.10.13
  • DataFrames v1.6.1
  • FileIO v1.16.3
  • GLMakie v0.9.9

Installation

You can install Porta by running this (in the REPL):

]add Porta

or,

Pkg.add("Porta")

or get the latest experimental code.

]add https://github.com/iamazadi/Porta.jl.git

Usage

For client-side code read the tests, and for examples on how to build, please check out the models directory. See planethopf.jl as an example.

Status

  • Logic [Doing]
  • Set Theory [TODO]
  • Topology [TODO]
  • Topological Manifolds [TODO]
  • Differentiable Manifolds [TODO]
  • Bundles [TODO]
  • Geometry: Symplectic, Metric [TODO]
  • Documentation [TODO]
  • Geometric Unity [TODO]

References

+Home · Porta.jl
GitHub

Geometrize the quantum!

This project is inspired by Eric Weinstein's Graph-Wall-Tome (GWT) project. Watch visual models on the YouTube channel.

Requirements

  • CSV v0.10.13
  • DataFrames v1.6.1
  • FileIO v1.16.3
  • GLMakie v0.9.9

Installation

You can install Porta by running this (in the REPL):

]add Porta

or,

Pkg.add("Porta")

or get the latest experimental code.

]add https://github.com/iamazadi/Porta.jl.git

Usage

For client-side code read the tests, and for examples on how to build, please check out the models directory. See planethopf.jl as an example.

Status

  • Logic [Doing]
  • Set Theory [TODO]
  • Topology [TODO]
  • Topological Manifolds [TODO]
  • Differentiable Manifolds [TODO]
  • Bundles [TODO]
  • Geometry: Symplectic, Metric [TODO]
  • Documentation [TODO]
  • Geometric Unity [TODO]

References

diff --git a/dev/newsreport.html b/dev/newsreport.html index e4489dc..0d96fbf 100644 --- a/dev/newsreport.html +++ b/dev/newsreport.html @@ -59,4 +59,13 @@ axis = normalize(ℝ³(vec(p)[2:4])) M = mat4(Quaternion(progress * 4π, axis)) ο_transformed = M * Quaternion(vec(ο)) -ι_transformed = M * Quaternion(vec(ι))

innerproductspositivechina

For example, the Standard Model is formulated on 4-dimensional Minkowski spacetime, over which all fiber bundles can be trivialized and spinors have a simple explicit description. For the Symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In theoretical physics, Lie groups like the Lorentz and Poincaré groups, which are related to spacetime symmetries, and gauge groups, defining internal symmetries, are important cornerstones. Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group. The Lie algebra $SL(2,\mathbb{C})$ plays a special role in physics, because as a real Lie algebra it is isomorphic to the Lie algebra of the Lorentz group of 4-dimensional spacetime. At least locally, fields in physics can be described by maps on spacetime with values in vector spaces.

The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. As we will discuss in Sect. 6.8.2 in more detail, the group $SL(2,\mathbb{C})$ is the (orthochronous) Lorentz spin group, i.e. the universal covering of the identity component of the Lorentz group of 4-dimensional spacetime. The fundamental geometric opbject in a gauge theory is a principal bundle over spacetime with structure group given by the gauge group. The fibers of a principal bundle are sometimes thought of as an internal space at every spacetime point, not belonging to spacetime itself. Fiber bundles are indispensible in gauge theory and physics in the situation where spacetime, the base manifold, has a non-trivial topology.

It also happens if we compactify (Euclidean) spacetime $\mathbb{R}^4$ to the 4-sphere $S^4$. In these situations, fields on spacetime often cannot be described simply by a map to a fixed vector space, but rather as sections of a non-trivial vector bundle. We will see that this is similar to the difference in special relativity between Minkowski spacetime and the choice of an inertial system. This can be compared, in special relativity, to the choice of an inertial system for Minkowski spacetime $M$, which defines an identification on $M \cong \mathbb{R}^4$. Of course, different choices of gauges are possible, leading to different trivializations of the principal bundle, just as different choices of inertial systems lead to different identifications of spacetime with $\mathbb{R}^4$.

Note that, if we consider principal bundles over Minkowski spacetime $\mathbb{R}^4$, it does not matter for this discussion that principal bundles over Euclidean spaces are always trivial by Corollary 4.2.9. This is very similar to special relativity, where spacetime is trivial, i.e. isomorphic to $\mathbb{R}^4$ with a Minkowski metric, but what matters is the independence of the actual trivialization, i.e. the choice of inertial system. Table 4.2 Comparison between notions for special relativity and gauge theory

ManifoldTrivializationTransformations and invariance
Special relativitySpacetime $M$$M \cong \mathbb{R}^4$ via inertial systemLorentz
Gauge theoryPrincipal bundle $P \to M$$P \cong M \times G$ via choice of gaugeGauge

It follows that, given a local gauge of the gauge bundle $P$, the section in $E$ corresponds to a unique local map from spacetime into the vector space $V$. In particular, we can describe matter fields on a spacetime diffeomorphic to $\mathbb{R}^4$ by unique maps from $\mathbb{R}^4$ into a vector space, once a global gauge for the principal bundle has been chosen. At least locally (after a choice of local gauge) we can interpret connection 1-forms as fields on spacetime (the base manifold) with values in the Lie algebra of the gauge group. Notice that connections are not unique (if $dim M, dim G \ge 1$), not even in the case of trivial principal bundles (all connections that appear in the Standard Model over Minkowski spacetime, for example, are defined on trivial principal bundles). The diffeomorphism group $Diff(M)$ of spacetime $M$ plays a comparable role in general relativity.

This is related to the fact that gauge theories describe local interactions (the interactions occur in single spacetime points). The local connection 1-form is thus defined on an open subset in the base manifold $M$ and can be considered as a "field on spacetime" in the usual sense. Generalized Electric and Magnetic Fields on Minkowski Spacetime of Dimension 4 In quantum field theory, the gauge field $A_{\mu}$ is a function on spacetime with values in the operators on the Hilbert state space $V$ (if we ignore for the moment questions of whether this operator is well-defined and issues of regularization). By Corollary 5.13.5 this difference can be identified with a 1-form on spacetime $M$ with values in $Ad(P)$.

In physics this fact is expressed by saying that gauge bosons, the differences $A_{\mu}-A_{\mu}^0$, are fields on spacetime that transform in the adjoint representation of $G$ under gauge transformations. In the case of Minkowski spacetime, rotations correspond to Lorentz transformations. The pseudo-Riemannian case, like the case of Minkowski spacetime, is discussed less often, even though it is very important for physics (a notable exception is the thorough discussion in Helga Baun's book [13]). $\mathbb{R}^{s,1}$ and $\mathbb{R}^{1,t}$ are the two versions of Minkowski spacetime (both versions are used in physics). This includes the particular case of the Lorentz group of Minkowski spacetime.

However, as mentioned above, depending on the convention, 4-dimensional Minkowski spacetime in quantum field theory can have signature $(+,-,-,-)$, so that time carries the plus sign. Example 6.1.20 For applications concerning the Standard Model, the most important of these groups is the proper orthochronous Lorentz group $SO^+(1,3) \cong SO^+(3,1)$ of 4-dimensional Minkoeski spacetime. They are physical gamma matrices for $Cl(1,3)$, i.e. for the Clifford algebra of Minkowski spacetime with signature $(+,-,-,-)$, in the so-called Weyl representation or chiral representation. Example 6.3.18 Let $\Gamma_a$ and $\gamma_a = i \Gamma_a$ be the physical and mathematical gamma matrices for $Cl(1,3)$ considered in Example 6.3.17. If we set $\Gamma_a^\prime = \gamma_a$, $\gamma_a^\prime = i \Gamma_a^\prime = -\Gamma_a$, then these are physical and Mathematical gamma matrices for $Cl(1,3)$ of Minkowski spacetime with signature $(-,+,+,+)$. Example 6.3.24 For Minkowski spacetime of dimension 4 we have Table 6.1 Complex Clifford algebras

$n$$Cl(n)$$Cl^0(n)$$N$
Evan$End(\mathbb{C}^N)$$End(\mathbb{C}^{N/2}) \oplus End(\mathbb{C}^{N/2})$$2^{n/2}$
Odd$End(\mathbb{C}^N) \oplus End(\mathbb{C}^N)$$End(\mathbb{C}^N)$$2^{(n-1)/2}$

Table 6.2 Real Clifford algebras

$\rho \ mod \ 8$$Cl(s,t)$$N$
$0$$End(\mathbb{R}^N)$$2^{n/2}$
$1$$End(\mathbb{C}^N)$$2^{(n-1)/2}$
$2$$End(\mathbb{H}^N)$$2^{(n-2)/2}$
$3$$End(\mathbb{H}^N) \oplus End(\mathbb{H}^N)$$2^{(n-3)/2}$
$4$$End(\mathbb{H}^N)$$2^{(n-2)/2}$
$5$$End(\mathbb{C}^N)$$2^{(n-1)/2}$
$6$$End(\mathbb{R})$$2^{n/2}$
$7$$End(\mathbb{R}^N) \oplus End(\mathbb{R}^N)$$2^{(n-1)/2}$

Table 6.3 Even part of real Clifford algebras

$\rho \ mod \ 8$$Cl^0(s,t)$$N$
$0$$End(\mathbb{R}^N) \oplus End(\mathbb{R}^N)$$2^{(n-2)/2}$
$1$$End(\mathbb{R}^N)$$2^{(n-1)/2}$
$2$$End(\mathbb{C}^N)$$2^{(n-2)/2}$
$3$$End(\mathbb{H}^N)$$2^{(n-3)/2}$
$4$$End(\mathbb{H}^N) \oplus End(\mathbb{H}^N)$$2^{(n-4)/2}$
$5$$End(\mathbb{H}^N)$$2^{(n-3)/2}$
$6$$End(\mathbb{C^N})$$2^{(n-2)/2}$
$7$$End(\mathbb{R}^N)$$2^{(n-1)/2}$

$Cl(1,3) \cong End(\mathbb{R}^4)$

$Cl(3,1) \cong End(\mathbb{H}^2)$

$Cl^0(1,3) \cong Cl^0(3,1) \cong End(\mathbb{C}^2)$

Example 6.6.7 For Minkowski spacetime $\mathbb{R}^{n-1,1}$ of dimension $n$ we have $n = \rho + 2$.

We see that in Minkowski spacetime of dimension 4 there exist both Majorana and Weyl spinors of real dimension 4, but not Majorana-Weyl spinors. In quantum field theory, spinors become fields of operators on spacetime acting on a Hilbert space. Explicit formulas for Minkowski Spacetime of Dimension 4 We collect some explicit formulas concerning Clifford algebras and spinors for the case of 4-dimensional Minkowski spacetime. In Minkowski spacetime of dimension 4 and signature $(+,-,-,-)$ (usually used in quantum field theory) there exist both Weyl and Majorana spinors, but not Majorana-Weyl spinors. Our aim in this subsection is to prove that the orthochronous spin group $Spin^+(1,3)$ of 4-dimensional Minkowski spacetime is isomorphic to the 6-dimensional Lie group $SL(2,\mathbb{C})$.

The Story

Who

With the discovery of a new particle, announced on 4 July 2012 at CERN, whose properties are "consistent with the long-sought Higgs boson" [31], the final elementary particle predicted by the classical Standard Model of particle physics has been found. Interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs bosons), determined by the Lagrangian. The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. The fact that there are 8 gluons, 3 weak gauge bosons, and 1 photon is related to the dimensions of the Lie groups $SU(3)$ and $SU(2) \times U(1)$. Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group.

The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields. The gauge bosons corresponding to these gauge groups are described by the adjoint representation that we discuss in Sect. 2.1.5. The representation $Ad_H$ describes the representation of the gauge boson fields in the Standard Model. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]).

Connections on principal bundles, that we discuss in Chap. 5, correspond to gauge fields whose particle excitations in the associated quantum field theory are the gauge bosons that transmit interactions. These fields are often called gauge fields and correspond in the associated quatum field theory to gauge bosons. This implies a direct interaction between gauge bosons (the gluons in QCD) that does not occur in abelian gauge theories like quantum electrodynamics (QED). The difficulties that are still present nowadays in trying to understand the quantum version of non-abelian gauge theories, like quantum chromodynamics, can ultimately be traced back to this interaction between gauge bosons. The real-valued fields $A_\mu^a \in C^\infty(U,\mathbb{R})$ and the corresponding real-valued 1-forms $A_s \in \Omega^1(U)$ are called (local) gauge boson fields.

In physics, the quadratic term $[A_\mu, A_\nu]$ in the expression for $F_{\mu\nu}$ (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field $A_\mu$. This explains why gluons, the gauge bosons of QCD, interact directly with each other, while photons, the gauge bosons of QED, do not. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by $A_\mu$ and the particles described by the field $\phi$. We then get a better understanding of why gauge bosons in physics are said to transform under the adjoint representation.

Strictly speaking, gauge bosons, the excitations of the gauge field, should then be described classically by the difference $A - A^0$, where $A$ is some other connection 1-form and not by the field $A$ itself. In physics this fact is expressed by saying that gauge bosons, the differences $A_\mu - A_\mu^0$, are fields on spacetime that transform in the adjoint represntation of $G$ under gauge transformations. Gauge fields correspond to gauge bosons (spin 1 particles) and are described by 1-forms or, dually, vector fields. Even though spinors are elementary objects, some of their properties (like the periodicity modulo 8, real and quaternionic structures, or bilinear and Hamiltonian scalar products) are not at all obvious, already on the level of linear algebra, and do not have a direct analogue in the bosonic world of vectors and tensors. The existence of gauge symmetries is particularly important: it can be shown that a quantum field theory involving massless spin 1 bosons can be consistent (i.e. unitary, see Sect. 7.1.3) only if it is gauge invariant [125,143].

Graph

graph

What

The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. Spin groups such as the universal covering of the Lorentz group and its higher dimesnional analogues, are also important in physics, because they are involved in the mathematical description of fermions. Counting in this way, the Standard Model thus contains at the most elementary level 90 fermions (particles and antiparticles). The complex vector space $V$ of fermions, which carries a representation of $G$, has dimension 45 (plus the same number of corresponding antiparticles) and is the direct sum of the two G-invariant subspaces (sectors): a lepton sector of dimension 9 (where we do not include the hypothetical right-handed neutrinos) and a quark sector of dimension 36. Matter fields in the Standard Model, like quarks and leptons, or sacalar fields, like the Higgs field, correspond to sections of vector bundles associated to the principal bundle (and twisted by spinor bundles in the case of fermions).

For example, in the Standard Model, one generation of fermions is described by associated complex vector bundles of rank 8 for left-handed fermions and rank 7 for right-handed fermions, associated to representations of the gauge group $SU(3) \times SU(2) \times U(1)$. Matter fields in physics are described by smooth sections of vector bundles $E$ associated to principal bundles $P$ via the representations of the gauge group $G$ on a vector space $V$ (in the case of fermions the associated bundle $E$ is twisted in addition with a spinor bundle $S$, i.e. the bundle is $S \otimes E$). Additional matter fields, like fermions or scalars, can be introduced using associated vector bundles. These particles are fermions (spin $\frac{1}{2}$ particles) and are described by spinor fields (spinors). Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly neutrinos) and, together with the Dirac operator, the kinetic term and the interaction term; see Sect. 7.6. This is related to the fact that the weak interaction in the Standard Model is not invariant under parity inversion that exchanges left-handed with right-handed fermions.

Perspective

How

Hence, by the uniqueness of integral curves (which is a theorem about the uniqueness of solutions to odrinary differential equations) we have $\phi_X(s) \cdot \phi_X(t) = \phi_X(s + t) \ \forall t \in I \cap (t_{min} - s, t_{max} - s)$. This implies the claim by uniqueness of solutions of ordinary differential equations. The unique solution of this differential equation for $\gamma(t)$ is $\gamma(t) = e^{tr(X)t}$. Then $e^D = \begin{bmatrix} e^{d_1} & 0 & 0 \\ 0 & e^{d_2} & 0 \\ & \ddots & \\ 0 & 0 & e^{d_n} \end{bmatrix}$ and the equation $det(e^D) = e^{d_1} ... e^{d_n} = e^{d_1 + ... + d_n} = e^{tr(D)}$ is trivially satisfied. Then we can calculate: $(R^*_gs)_p(X,Y) = \ <L_{(pg)^{-1}*}R_{g*}(X), L_{(pg)^{-1}*}R_{g*}(Y)> \ = \ <Ad_{g^{-1}} \circ L_{p^{-1}*}(X), Ad_{g^{-1}} \circ L_{p^{-1}*}(Y)>$ and $s_p(X,Y) \ = \ <L_{p^{-1}*}(X), L_{p^{-1}*}(Y)>$, where in both equations we used that $s$ is left invariant.

Lemma 3.3.3 For $A \in Mat(m \times m, \mathbb{H})$ and $v \in \mathbb{H}^m$ the following equation holds: $det\begin{bmatrix}1 & v \\ 0 & A\end{bmatrix} = det(A)$. Lemma 4.1.13 (Cocycle Conditions) The transition functions $\{\phi_{ij}\}_{i,j \in I}$ satisfy the following equations:

The third equation is called the cycycle condition.

5.5.2 The structure equation Theorem 5.5.4 (Structure Equation) The curvature form $F$ of a connection form $A$ satisfies $F = dA + \frac{1}{2}[A,A]$.

Proof We check the formula by inserting $X,Y \in T_pP$ on both sides of the equation, where we distinguish the following three cases:

  1. Both $X$ and $Y$ are vertical: Then $X$ and $Y$ are fundamental vectors, $X = \tilde{V}_p, \ Y = \tilde{W}_p$ for certain elements $V,W \in g$. We get $F(X,Y) = dA(\pi_H(X), \pi_H(Y)) = 0$. On the other hand we have $\frac{1}{2}[A,A](X,Y) = [A(X),A(Y)] = [V,W]$. The differential $dA$ of a 1-form $A$ is given according to Proposition A.2.22 by $dA(X,Y) = L_X(A(Y))-L_Y(A(X))-A([X,Y])$, where we extend the vectors $X$ and $Y$ to vector fields in a neighbourhood of $p$. If we choose the extension by fundamental vector fields $\tilde{V}$ and $\tilde{W}$, then $dA(X,Y) = L_X(W) - L_Y(V) - [V,W] = -[V,W]$ since $V$ and $W$ are constant maps from $P$ to $g$ and we used that $[\tilde{V},\tilde{W}] = \tilde{[V,W]}$ according to Proposition 3.4.4. This implies the claim.
  2. Both $X$ and $Y$ are horizontal: Then $F(X,Y) = dA(X,Y)$ and $\frac{1}{2}[A,A](X,Y) = [A(X), A(Y)] = [0,0]=0$. This implies the claim.
  3. $X$ is vertical and $Y$ is horizontal: Then $X = \tilde{V}_p$ for some $V \in g$. We have $F(X,Y) = dA(\pi_H(X),\pi_H(Y)) = dA(0, Y) = 0$ and $\frac{1}{2}[A,A](X,Y) = [A(X),A(Y)] - [V,0] = 0$. Furthermore, $dA(X,Y) = L_{\tilde{V}}(A(Y)) - L_Y(V) - A([\tilde{V},Y]) = -A([\tilde{V},Y]) = 0$ since $[\tilde{V},Y]$ is horizontal by Lemma 5.5.5. This implies the claim.

The structure equation is very useful when we want to calculate the curvature of a given connection.

By the structure equation we have $F = dA + \frac{1}{2} [A, A]$ so that $dF = \frac{1}{2} d[A, A]$.

Proposition 5.6.2 (Local Structure Equation) The local field strength can be calculated as $F_s = dA_s + \frac{1}{2}[A_s,A_s]$ and $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]$.

It remains to check that $F_M$ is closed. In a local gauge $s$ we have according to the local structure equation $F_s = dA_s + \frac{1}{2}[A_s,A_s]$.

Proposition 5.6.8 For the connection on the Hopf bundle the following equation holds: $\frac{1}{2\pi i} \int_{S^2} F_{S^2} = 1$.

We write $A_\mu = A_s(\partial_\mu), F_{\mu\nu} = F_s(\partial_\mu, \partial_\nu)$ and we have the local structure equation $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]$.

We will determine $g(t)$ as the solution of a differential equation.

Proof Properties 1-3 follow from the theory of ordinary differential equations. (Parallel transport)

These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories.

Recall that for the proof of Theorem 5.8.2 concerning the existence of a horizontal lift $\gamma^*$ of a curve $\gamma:[0,1] \to M$ where $\gamma^*(0) = p \in P_{\gamma(0)}$, we had to solve the differential equation $\dot{g}(t) = -R_{g(t)*} A(\dot{\delta}(t))$, with $g(0) = e$, where $\delta$ is some lift of $\gamma$ and $g:[0,1] \to G$ is a map with $\gamma^*(t) = \delta(t) \cdot g(t)$.

Then the differential equation can be written as $\frac{dg(t)}{dt} = -A_s(\dot{\gamma}(t)) \cdot g(t)$.

Path-ordered exponentials are useful, because they define solutions to the ordinary differential equation we are interested in.

Then uniqueness of the solution to ordinary differential equations show that $g \equiv h$, hence $g$ takes values in $G$.

The solution to this differential equation is

$g(t) = P exp(- \int_0^t \sum_{\mu=1}^n A_{s\mu}(\gamma(s))\frac{dx^\mu}{ds}ds) = P exp(- \int_{\gamma(0)}^{\gamma(t)} \sum_{\mu=1}^n A_{s\mu} (x^\mu) dx^\mu) = P exp(- \int_{\gamma_t} A_s)$, where $\gamma_t$ denotes the restriction of the curve $\gamma$ to $[0,t]$.

What is the interpretation of the structure equation?

Taking the determinant of both sides of this equation shows that:

Lemma 6.1.7 Matrices $A \in O(s,t)$ satisfy $detA = \pm 1$.

$A^T \begin{bmatrix} I_s & 0 \\ 0 & -I_t \end{bmatrix} A = \begin{bmatrix} I_s & 0 \\ 0 & -I_t \end{bmatrix}$.

Remark 6.2.5 We can think of the linear map $\gamma$ as a linear square root of the symmetric bilinear form $-Q$: in the definition of Clifford algebras, it suffices to demand that $\gamma(v)^2 = -Q(v,v) \cdot 1 \ \forall \ v, w \in V$, because, considering this equation for vectors $v, w, v + w$, the equation $\{\gamma(v), \gamma(w\} = -2Q(v, w) \cdot 1 \ \forall \ v, w \in V$ follows.

Lemma 6.3.6 Every chirality element $\omega$ satisfies

Proof The first equation follows from $e_a \cdot \omega = \lambda e_a \cdot e_1 ... e_n = (-1)^{a - 1} \lambda e_1 ... e_a \cdot e_a ... e_n$

$\omega \cdot e_a = \lambda e_1 ... e_n \cdot e_a = (-1)^{n - a} \lambda e_1 ... e_a \cdot e_a ... e_n = -e_a \cdot \omega$,

since $n$ is even. The second equation is a consequence of the first.

Let $\Gamma_1, ..., \Gamma_n$ be physical gamma matrices. We set

$\Gamma_a = \eta^{ac} \Gamma_c$,

$\Gamma^{bc} = \frac{1}{2} [\Gamma^b, \Gamma^c] = \frac{1}{2} (\Gamma^b \Gamma^c - \Gamma^c \Gamma^b)$,

$\Gamma^{n + 1} = -i^{k + t} \Gamma^1 ... \Gamma^n$

and similarly for the mathematical $\gamma$-matrices (in the first equation there is an implicit sum over $c$; this is an instance of the Einstein summation convention). These matrices satisfy by Lemma 6.3.6

$\{\Gamma^{n + 1}, \Gamma^a\} = 0$,

$[\Gamma^{n + 1}, \Gamma^{bc}] = 0$,

$\gamma^{bc} = -\Gamma^{bc}$.

In the following examples we use the Pauli matrices

$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $\sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$, $\sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$.

It is easy to check that they satisfy the identities

$\sigma^2 = I_2 \ \ j = 1, 2, 3$,

$\sigma_j \sigma_{j + 1} = -\sigma_{j + 1} \sigma_{j} = i \sigma_{j + 2} \ \ j = 1, 2, 3$,

where in the second equation ${j + 1}$ and $j + 2$ are taken $mod 3$.

$(\psi, \phi) = \psi^T C \phi$

Furthermore property 1. and 2. in Definition 6.7.1 are equivalent to

  1. $\gamma_a^T = \mu C \gamma_a C^{-1} \ \ for \ all \ a = 1, ..., s + t$.
  2. $C^T = \nu C$.

The first equation also holds with the physical Clifford matrices $\Gamma_a$ instead of the mathematical matrices $\gamma_a$.

There is an equivalent equation to the first one with physical Clifford matrices $\Gamma_a : 1 \cdot \Gamma^\dagger_a = -\delta A \Gamma_a A^{-1} \ \ for \ all \ a = 1, ..., s + t$.

Furthermore, property 1. and 2. in Definition 6.7.8 are equivalent to:

  1. $\gamma_a^{\dagger} = \delta A \gamma_a A^{-1} \ \ for \ all \ a = 1, ..., s + t$.
  2. $A^{\dagger} = A$.

Given a spin structure on a pseudo-Riemannian manifold and the spinor bundle $S$, we would like to have a covariant derivative on $S$ so that we can define field equations involving derivatives of spinors.

The Iconic Wall

corrected-wall

Tome

tome

Wrap Up

Why

The following three chapters discuss applications in physics: the Lagrangians and interactions in the Standard Model, spontaneous symmetry breaking, the Higgs mechanism of mass generation, and some more advanced and modern topics like neutrino masses and CP violation. Depending on the time, the interests and the prior knowledge of the reader, he or she can take a shortcut and immediately start at the chapters on connections, spinors or Lagrangians, and then go back if more detailed mathematical knowledge is required at some point. An interesting and perhaps underappreciated fact is that a substantial number of phenomena in particle physics can be understood by analysing representations of Lie groups and by rewriting or rearranging Lagrangians.

Symmetries of Lagrangians interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs boson), determined by the Lagrangian. For the symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In the following chapter we will study some associated concepts, like representations (which are used to define the actions of Lie groups on fields) and invariant matrices (which are important in the construction of the gauge invariant Yang-Mills Lagrangian). We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields.

The existence of positive definite Ad-invariant scalar products on the Lie algebra of compact Lie groups is very important in gauge theory, in particular, for the construction of the gauge-invariant Yang-Mills Lagrangian; see Sect. 7.3.1. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]). In a gauge-invariant Lagrangian this results in terms of order higher than two in the matter and gauge fields, which are interpreted as interactions between the corresponding particles. In non-abelian gauge theories, like quantum chromodynamics (QCD), there are also terms in the Lagrangian of order higher than two in the gauge fields themselves, coming from a quadratic term in the curvature that appears in the Yang-Mills Lagrangian.

In physics, the quadratic term $[A_\mu, A_\nu]$ in the expression for $F_{\mu\nu}$ (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field $A_\mu$. These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by $A_\mu$ and the particles described by the field $\phi$.

feynmandiagrams

Figure 5.2 shows the Feynman diagrams for the cubic and quartic terms which appear in the Klein-Gordon Lagrangian in Eq. (7.3), representing the interaction between a gauge field $A$ and a charged scalar field described locally by a map $\phi$ with values in $V$. Fig 5.2 Feynman diagrams for interaction between gauge field and charged scalar

Hermitian scalar products are particularly important, because we need them in Chap. 7 to define Lorentz invariant Lagrangians involving spinors.

$<\psi, \phi> \ = \ \overline{\psi} \phi$,

$\overline{\psi} = \psi^\dagger A$.

Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly the neutrinos) and, together with the Dirac operator the kinetic term and the interaction term; see Sect. 7.6.

Porta.jl

References

  1. Mark J.D. Hamilton, Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics, Springer Cham, DOI, published: 10 January 2018.

  2. Sir Roger Penrose, The Road to Reality, (2004).

  3. Roger Penrose, Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).

  4. Richard M. Murray and Zexiang Li, A Mathematical Introduction to Robotic Manipulation, 1st Edition, 1994, CRC Press, read, buy.

  5. Edward Witten, Physics and Geometry, (1987).

  6. The iconic Wall of Stony Brook University.

+ι_transformed = M * Quaternion(vec(ι))

innerproductspositivechina

ζ = Complex(_κ + _ω)
+_τ = SpinVector(ζ, timesign)
+ζ′ = Complex(_κ′ + _ω′)
+_τ′ = SpinVector(ζ′, timesign)
+gauge1 = -imag(dot(_κ, _ω))
+gauge2 = -imag(dot(_κ, _τ))
+gauge3 = float(π)
+@assert(isapprox(dot(_τ, _ι), vec(_τ)[1]), "The second component of the spin vector $_τ  is not equal to minus the inner product of $_τ and $_ι.")
+@assert(isapprox(dot(_τ, _ο), -vec(_τ)[2]), "The second component of the spin vector $_τ is not equal to minus the inner product of $_τ and $_ο.")

The geometry of "spin-vector addition" is shown. The spin-vectors exist in a spin-space that is equipped with three operations: scalar multiplication, inner product and addition. The addition of spin-vectors κ and ω results in another spin-vector κ + ω in the spin-space, which has its own flagpole and flag plane. Taking κ and ω as null vectors in the sphere of future null directions, the flagpole of κ is represented by a point (complex number) and the null flag of κ is represented as a point sufficiently close to κ that is used to assign a direction tangent to the sphere at κ.

addition02

The tails of the flagpoles of κ, ω and κ + ω are in a circle in the sphere of future null directions. The circumcircle of the triangle made by joining the tails of the three spin-vectors makes angles with the flagpoles and null planes. Meaning, the distance between κ and the center of the circle is equal to the distance between ω and the center. Also, the distance of the addition of κ and ω and the circle center is the same as the distance between κ and the center. For the circumcircle, we have three collinear points in the Argand complex plane. However, lines in the Argand plane become circles in sections of the three-dimensional sphere. The angle that the flagpoles of κ and ω make with the circle should be twice the argument of the inner product of the two spin-vectors (modulus 2π with a possible addition of π).

addition08

w = (Complex(κ + ω) - Complex(κ)) / (Complex(ω) - Complex(κ))
+@assert(imag(w) ≤ 0 || isapprox(imag(w), 0.0), "The flagpoles are not collinear: $(Complex(κ)), $(Complex(ω)), $(Complex(κ + ω))")

In an interesting way, the argument (phase) of the inner product of κ and ω is equal to half of the sum of the angles that the spin-vectors make with the circle, which is in turn equal to the angle that U and V make with each other minus π (also see the geometric descriptions of the inner product to construct U and V). In the case of spin-vector addition, the angles that the flag planes of κ, ω and κ + ω, each make with the circle are equal. But, be careful with determining the signs of the flag planes and the possible addition of π to the flag plane of κ + ω. For determining flag plane signs, see also Figure 1-21 in page 64 of Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).

addition09

For example, the Standard Model is formulated on 4-dimensional Minkowski spacetime, over which all fiber bundles can be trivialized and spinors have a simple explicit description. For the Symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In theoretical physics, Lie groups like the Lorentz and Poincaré groups, which are related to spacetime symmetries, and gauge groups, defining internal symmetries, are important cornerstones. Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group. The Lie algebra $SL(2,\mathbb{C})$ plays a special role in physics, because as a real Lie algebra it is isomorphic to the Lie algebra of the Lorentz group of 4-dimensional spacetime. At least locally, fields in physics can be described by maps on spacetime with values in vector spaces.

The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. As we will discuss in Sect. 6.8.2 in more detail, the group $SL(2,\mathbb{C})$ is the (orthochronous) Lorentz spin group, i.e. the universal covering of the identity component of the Lorentz group of 4-dimensional spacetime. The fundamental geometric opbject in a gauge theory is a principal bundle over spacetime with structure group given by the gauge group. The fibers of a principal bundle are sometimes thought of as an internal space at every spacetime point, not belonging to spacetime itself. Fiber bundles are indispensible in gauge theory and physics in the situation where spacetime, the base manifold, has a non-trivial topology.

It also happens if we compactify (Euclidean) spacetime $\mathbb{R}^4$ to the 4-sphere $S^4$. In these situations, fields on spacetime often cannot be described simply by a map to a fixed vector space, but rather as sections of a non-trivial vector bundle. We will see that this is similar to the difference in special relativity between Minkowski spacetime and the choice of an inertial system. This can be compared, in special relativity, to the choice of an inertial system for Minkowski spacetime $M$, which defines an identification on $M \cong \mathbb{R}^4$. Of course, different choices of gauges are possible, leading to different trivializations of the principal bundle, just as different choices of inertial systems lead to different identifications of spacetime with $\mathbb{R}^4$.

Note that, if we consider principal bundles over Minkowski spacetime $\mathbb{R}^4$, it does not matter for this discussion that principal bundles over Euclidean spaces are always trivial by Corollary 4.2.9. This is very similar to special relativity, where spacetime is trivial, i.e. isomorphic to $\mathbb{R}^4$ with a Minkowski metric, but what matters is the independence of the actual trivialization, i.e. the choice of inertial system. Table 4.2 Comparison between notions for special relativity and gauge theory

ManifoldTrivializationTransformations and invariance
Special relativitySpacetime $M$$M \cong \mathbb{R}^4$ via inertial systemLorentz
Gauge theoryPrincipal bundle $P \to M$$P \cong M \times G$ via choice of gaugeGauge

It follows that, given a local gauge of the gauge bundle $P$, the section in $E$ corresponds to a unique local map from spacetime into the vector space $V$. In particular, we can describe matter fields on a spacetime diffeomorphic to $\mathbb{R}^4$ by unique maps from $\mathbb{R}^4$ into a vector space, once a global gauge for the principal bundle has been chosen. At least locally (after a choice of local gauge) we can interpret connection 1-forms as fields on spacetime (the base manifold) with values in the Lie algebra of the gauge group. Notice that connections are not unique (if $dim M, dim G \ge 1$), not even in the case of trivial principal bundles (all connections that appear in the Standard Model over Minkowski spacetime, for example, are defined on trivial principal bundles). The diffeomorphism group $Diff(M)$ of spacetime $M$ plays a comparable role in general relativity.

This is related to the fact that gauge theories describe local interactions (the interactions occur in single spacetime points). The local connection 1-form is thus defined on an open subset in the base manifold $M$ and can be considered as a "field on spacetime" in the usual sense. Generalized Electric and Magnetic Fields on Minkowski Spacetime of Dimension 4 In quantum field theory, the gauge field $A_{\mu}$ is a function on spacetime with values in the operators on the Hilbert state space $V$ (if we ignore for the moment questions of whether this operator is well-defined and issues of regularization). By Corollary 5.13.5 this difference can be identified with a 1-form on spacetime $M$ with values in $Ad(P)$.

In physics this fact is expressed by saying that gauge bosons, the differences $A_{\mu}-A_{\mu}^0$, are fields on spacetime that transform in the adjoint representation of $G$ under gauge transformations. In the case of Minkowski spacetime, rotations correspond to Lorentz transformations. The pseudo-Riemannian case, like the case of Minkowski spacetime, is discussed less often, even though it is very important for physics (a notable exception is the thorough discussion in Helga Baun's book [13]). $\mathbb{R}^{s,1}$ and $\mathbb{R}^{1,t}$ are the two versions of Minkowski spacetime (both versions are used in physics). This includes the particular case of the Lorentz group of Minkowski spacetime.

However, as mentioned above, depending on the convention, 4-dimensional Minkowski spacetime in quantum field theory can have signature $(+,-,-,-)$, so that time carries the plus sign. Example 6.1.20 For applications concerning the Standard Model, the most important of these groups is the proper orthochronous Lorentz group $SO^+(1,3) \cong SO^+(3,1)$ of 4-dimensional Minkoeski spacetime. They are physical gamma matrices for $Cl(1,3)$, i.e. for the Clifford algebra of Minkowski spacetime with signature $(+,-,-,-)$, in the so-called Weyl representation or chiral representation. Example 6.3.18 Let $\Gamma_a$ and $\gamma_a = i \Gamma_a$ be the physical and mathematical gamma matrices for $Cl(1,3)$ considered in Example 6.3.17. If we set $\Gamma_a^\prime = \gamma_a$, $\gamma_a^\prime = i \Gamma_a^\prime = -\Gamma_a$, then these are physical and Mathematical gamma matrices for $Cl(1,3)$ of Minkowski spacetime with signature $(-,+,+,+)$. Example 6.3.24 For Minkowski spacetime of dimension 4 we have Table 6.1 Complex Clifford algebras

$n$$Cl(n)$$Cl^0(n)$$N$
Evan$End(\mathbb{C}^N)$$End(\mathbb{C}^{N/2}) \oplus End(\mathbb{C}^{N/2})$$2^{n/2}$
Odd$End(\mathbb{C}^N) \oplus End(\mathbb{C}^N)$$End(\mathbb{C}^N)$$2^{(n-1)/2}$

Table 6.2 Real Clifford algebras

$\rho \ mod \ 8$$Cl(s,t)$$N$
$0$$End(\mathbb{R}^N)$$2^{n/2}$
$1$$End(\mathbb{C}^N)$$2^{(n-1)/2}$
$2$$End(\mathbb{H}^N)$$2^{(n-2)/2}$
$3$$End(\mathbb{H}^N) \oplus End(\mathbb{H}^N)$$2^{(n-3)/2}$
$4$$End(\mathbb{H}^N)$$2^{(n-2)/2}$
$5$$End(\mathbb{C}^N)$$2^{(n-1)/2}$
$6$$End(\mathbb{R})$$2^{n/2}$
$7$$End(\mathbb{R}^N) \oplus End(\mathbb{R}^N)$$2^{(n-1)/2}$

Table 6.3 Even part of real Clifford algebras

$\rho \ mod \ 8$$Cl^0(s,t)$$N$
$0$$End(\mathbb{R}^N) \oplus End(\mathbb{R}^N)$$2^{(n-2)/2}$
$1$$End(\mathbb{R}^N)$$2^{(n-1)/2}$
$2$$End(\mathbb{C}^N)$$2^{(n-2)/2}$
$3$$End(\mathbb{H}^N)$$2^{(n-3)/2}$
$4$$End(\mathbb{H}^N) \oplus End(\mathbb{H}^N)$$2^{(n-4)/2}$
$5$$End(\mathbb{H}^N)$$2^{(n-3)/2}$
$6$$End(\mathbb{C^N})$$2^{(n-2)/2}$
$7$$End(\mathbb{R}^N)$$2^{(n-1)/2}$

$Cl(1,3) \cong End(\mathbb{R}^4)$

$Cl(3,1) \cong End(\mathbb{H}^2)$

$Cl^0(1,3) \cong Cl^0(3,1) \cong End(\mathbb{C}^2)$

Example 6.6.7 For Minkowski spacetime $\mathbb{R}^{n-1,1}$ of dimension $n$ we have $n = \rho + 2$.

We see that in Minkowski spacetime of dimension 4 there exist both Majorana and Weyl spinors of real dimension 4, but not Majorana-Weyl spinors. In quantum field theory, spinors become fields of operators on spacetime acting on a Hilbert space. Explicit formulas for Minkowski Spacetime of Dimension 4 We collect some explicit formulas concerning Clifford algebras and spinors for the case of 4-dimensional Minkowski spacetime. In Minkowski spacetime of dimension 4 and signature $(+,-,-,-)$ (usually used in quantum field theory) there exist both Weyl and Majorana spinors, but not Majorana-Weyl spinors. Our aim in this subsection is to prove that the orthochronous spin group $Spin^+(1,3)$ of 4-dimensional Minkowski spacetime is isomorphic to the 6-dimensional Lie group $SL(2,\mathbb{C})$.

The Story

Who

With the discovery of a new particle, announced on 4 July 2012 at CERN, whose properties are "consistent with the long-sought Higgs boson" [31], the final elementary particle predicted by the classical Standard Model of particle physics has been found. Interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs bosons), determined by the Lagrangian. The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. The fact that there are 8 gluons, 3 weak gauge bosons, and 1 photon is related to the dimensions of the Lie groups $SU(3)$ and $SU(2) \times U(1)$. Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group.

The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields. The gauge bosons corresponding to these gauge groups are described by the adjoint representation that we discuss in Sect. 2.1.5. The representation $Ad_H$ describes the representation of the gauge boson fields in the Standard Model. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]).

Connections on principal bundles, that we discuss in Chap. 5, correspond to gauge fields whose particle excitations in the associated quantum field theory are the gauge bosons that transmit interactions. These fields are often called gauge fields and correspond in the associated quatum field theory to gauge bosons. This implies a direct interaction between gauge bosons (the gluons in QCD) that does not occur in abelian gauge theories like quantum electrodynamics (QED). The difficulties that are still present nowadays in trying to understand the quantum version of non-abelian gauge theories, like quantum chromodynamics, can ultimately be traced back to this interaction between gauge bosons. The real-valued fields $A_\mu^a \in C^\infty(U,\mathbb{R})$ and the corresponding real-valued 1-forms $A_s \in \Omega^1(U)$ are called (local) gauge boson fields.

In physics, the quadratic term $[A_\mu, A_\nu]$ in the expression for $F_{\mu\nu}$ (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field $A_\mu$. This explains why gluons, the gauge bosons of QCD, interact directly with each other, while photons, the gauge bosons of QED, do not. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by $A_\mu$ and the particles described by the field $\phi$. We then get a better understanding of why gauge bosons in physics are said to transform under the adjoint representation.

Strictly speaking, gauge bosons, the excitations of the gauge field, should then be described classically by the difference $A - A^0$, where $A$ is some other connection 1-form and not by the field $A$ itself. In physics this fact is expressed by saying that gauge bosons, the differences $A_\mu - A_\mu^0$, are fields on spacetime that transform in the adjoint represntation of $G$ under gauge transformations. Gauge fields correspond to gauge bosons (spin 1 particles) and are described by 1-forms or, dually, vector fields. Even though spinors are elementary objects, some of their properties (like the periodicity modulo 8, real and quaternionic structures, or bilinear and Hamiltonian scalar products) are not at all obvious, already on the level of linear algebra, and do not have a direct analogue in the bosonic world of vectors and tensors. The existence of gauge symmetries is particularly important: it can be shown that a quantum field theory involving massless spin 1 bosons can be consistent (i.e. unitary, see Sect. 7.1.3) only if it is gauge invariant [125,143].

Graph

graph

What

The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. Spin groups such as the universal covering of the Lorentz group and its higher dimesnional analogues, are also important in physics, because they are involved in the mathematical description of fermions. Counting in this way, the Standard Model thus contains at the most elementary level 90 fermions (particles and antiparticles). The complex vector space $V$ of fermions, which carries a representation of $G$, has dimension 45 (plus the same number of corresponding antiparticles) and is the direct sum of the two G-invariant subspaces (sectors): a lepton sector of dimension 9 (where we do not include the hypothetical right-handed neutrinos) and a quark sector of dimension 36. Matter fields in the Standard Model, like quarks and leptons, or sacalar fields, like the Higgs field, correspond to sections of vector bundles associated to the principal bundle (and twisted by spinor bundles in the case of fermions).

For example, in the Standard Model, one generation of fermions is described by associated complex vector bundles of rank 8 for left-handed fermions and rank 7 for right-handed fermions, associated to representations of the gauge group $SU(3) \times SU(2) \times U(1)$. Matter fields in physics are described by smooth sections of vector bundles $E$ associated to principal bundles $P$ via the representations of the gauge group $G$ on a vector space $V$ (in the case of fermions the associated bundle $E$ is twisted in addition with a spinor bundle $S$, i.e. the bundle is $S \otimes E$). Additional matter fields, like fermions or scalars, can be introduced using associated vector bundles. These particles are fermions (spin $\frac{1}{2}$ particles) and are described by spinor fields (spinors). Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly neutrinos) and, together with the Dirac operator, the kinetic term and the interaction term; see Sect. 7.6. This is related to the fact that the weak interaction in the Standard Model is not invariant under parity inversion that exchanges left-handed with right-handed fermions.

Perspective

How

Hence, by the uniqueness of integral curves (which is a theorem about the uniqueness of solutions to odrinary differential equations) we have $\phi_X(s) \cdot \phi_X(t) = \phi_X(s + t) \ \forall t \in I \cap (t_{min} - s, t_{max} - s)$. This implies the claim by uniqueness of solutions of ordinary differential equations. The unique solution of this differential equation for $\gamma(t)$ is $\gamma(t) = e^{tr(X)t}$. Then $e^D = \begin{bmatrix} e^{d_1} & 0 & 0 \\ 0 & e^{d_2} & 0 \\ & \ddots & \\ 0 & 0 & e^{d_n} \end{bmatrix}$ and the equation $det(e^D) = e^{d_1} ... e^{d_n} = e^{d_1 + ... + d_n} = e^{tr(D)}$ is trivially satisfied. Then we can calculate: $(R^*_gs)_p(X,Y) = \ <L_{(pg)^{-1}*}R_{g*}(X), L_{(pg)^{-1}*}R_{g*}(Y)> \ = \ <Ad_{g^{-1}} \circ L_{p^{-1}*}(X), Ad_{g^{-1}} \circ L_{p^{-1}*}(Y)>$ and $s_p(X,Y) \ = \ <L_{p^{-1}*}(X), L_{p^{-1}*}(Y)>$, where in both equations we used that $s$ is left invariant.

Lemma 3.3.3 For $A \in Mat(m \times m, \mathbb{H})$ and $v \in \mathbb{H}^m$ the following equation holds: $det\begin{bmatrix}1 & v \\ 0 & A\end{bmatrix} = det(A)$. Lemma 4.1.13 (Cocycle Conditions) The transition functions $\{\phi_{ij}\}_{i,j \in I}$ satisfy the following equations:

The third equation is called the cycycle condition.

5.5.2 The structure equation Theorem 5.5.4 (Structure Equation) The curvature form $F$ of a connection form $A$ satisfies $F = dA + \frac{1}{2}[A,A]$.

Proof We check the formula by inserting $X,Y \in T_pP$ on both sides of the equation, where we distinguish the following three cases:

  1. Both $X$ and $Y$ are vertical: Then $X$ and $Y$ are fundamental vectors, $X = \tilde{V}_p, \ Y = \tilde{W}_p$ for certain elements $V,W \in g$. We get $F(X,Y) = dA(\pi_H(X), \pi_H(Y)) = 0$. On the other hand we have $\frac{1}{2}[A,A](X,Y) = [A(X),A(Y)] = [V,W]$. The differential $dA$ of a 1-form $A$ is given according to Proposition A.2.22 by $dA(X,Y) = L_X(A(Y))-L_Y(A(X))-A([X,Y])$, where we extend the vectors $X$ and $Y$ to vector fields in a neighbourhood of $p$. If we choose the extension by fundamental vector fields $\tilde{V}$ and $\tilde{W}$, then $dA(X,Y) = L_X(W) - L_Y(V) - [V,W] = -[V,W]$ since $V$ and $W$ are constant maps from $P$ to $g$ and we used that $[\tilde{V},\tilde{W}] = \tilde{[V,W]}$ according to Proposition 3.4.4. This implies the claim.
  2. Both $X$ and $Y$ are horizontal: Then $F(X,Y) = dA(X,Y)$ and $\frac{1}{2}[A,A](X,Y) = [A(X), A(Y)] = [0,0]=0$. This implies the claim.
  3. $X$ is vertical and $Y$ is horizontal: Then $X = \tilde{V}_p$ for some $V \in g$. We have $F(X,Y) = dA(\pi_H(X),\pi_H(Y)) = dA(0, Y) = 0$ and $\frac{1}{2}[A,A](X,Y) = [A(X),A(Y)] - [V,0] = 0$. Furthermore, $dA(X,Y) = L_{\tilde{V}}(A(Y)) - L_Y(V) - A([\tilde{V},Y]) = -A([\tilde{V},Y]) = 0$ since $[\tilde{V},Y]$ is horizontal by Lemma 5.5.5. This implies the claim.

The structure equation is very useful when we want to calculate the curvature of a given connection.

By the structure equation we have $F = dA + \frac{1}{2} [A, A]$ so that $dF = \frac{1}{2} d[A, A]$.

Proposition 5.6.2 (Local Structure Equation) The local field strength can be calculated as $F_s = dA_s + \frac{1}{2}[A_s,A_s]$ and $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]$.

It remains to check that $F_M$ is closed. In a local gauge $s$ we have according to the local structure equation $F_s = dA_s + \frac{1}{2}[A_s,A_s]$.

Proposition 5.6.8 For the connection on the Hopf bundle the following equation holds: $\frac{1}{2\pi i} \int_{S^2} F_{S^2} = 1$.

We write $A_\mu = A_s(\partial_\mu), F_{\mu\nu} = F_s(\partial_\mu, \partial_\nu)$ and we have the local structure equation $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + [A_\mu, A_\nu]$.

We will determine $g(t)$ as the solution of a differential equation.

Proof Properties 1-3 follow from the theory of ordinary differential equations. (Parallel transport)

These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories.

Recall that for the proof of Theorem 5.8.2 concerning the existence of a horizontal lift $\gamma^*$ of a curve $\gamma:[0,1] \to M$ where $\gamma^*(0) = p \in P_{\gamma(0)}$, we had to solve the differential equation $\dot{g}(t) = -R_{g(t)*} A(\dot{\delta}(t))$, with $g(0) = e$, where $\delta$ is some lift of $\gamma$ and $g:[0,1] \to G$ is a map with $\gamma^*(t) = \delta(t) \cdot g(t)$.

Then the differential equation can be written as $\frac{dg(t)}{dt} = -A_s(\dot{\gamma}(t)) \cdot g(t)$.

Path-ordered exponentials are useful, because they define solutions to the ordinary differential equation we are interested in.

Then uniqueness of the solution to ordinary differential equations show that $g \equiv h$, hence $g$ takes values in $G$.

The solution to this differential equation is

$g(t) = P exp(- \int_0^t \sum_{\mu=1}^n A_{s\mu}(\gamma(s))\frac{dx^\mu}{ds}ds) = P exp(- \int_{\gamma(0)}^{\gamma(t)} \sum_{\mu=1}^n A_{s\mu} (x^\mu) dx^\mu) = P exp(- \int_{\gamma_t} A_s)$, where $\gamma_t$ denotes the restriction of the curve $\gamma$ to $[0,t]$.

What is the interpretation of the structure equation?

Taking the determinant of both sides of this equation shows that:

Lemma 6.1.7 Matrices $A \in O(s,t)$ satisfy $detA = \pm 1$.

$A^T \begin{bmatrix} I_s & 0 \\ 0 & -I_t \end{bmatrix} A = \begin{bmatrix} I_s & 0 \\ 0 & -I_t \end{bmatrix}$.

Remark 6.2.5 We can think of the linear map $\gamma$ as a linear square root of the symmetric bilinear form $-Q$: in the definition of Clifford algebras, it suffices to demand that $\gamma(v)^2 = -Q(v,v) \cdot 1 \ \forall \ v, w \in V$, because, considering this equation for vectors $v, w, v + w$, the equation $\{\gamma(v), \gamma(w\} = -2Q(v, w) \cdot 1 \ \forall \ v, w \in V$ follows.

Lemma 6.3.6 Every chirality element $\omega$ satisfies

Proof The first equation follows from $e_a \cdot \omega = \lambda e_a \cdot e_1 ... e_n = (-1)^{a - 1} \lambda e_1 ... e_a \cdot e_a ... e_n$

$\omega \cdot e_a = \lambda e_1 ... e_n \cdot e_a = (-1)^{n - a} \lambda e_1 ... e_a \cdot e_a ... e_n = -e_a \cdot \omega$,

since $n$ is even. The second equation is a consequence of the first.

Let $\Gamma_1, ..., \Gamma_n$ be physical gamma matrices. We set

$\Gamma_a = \eta^{ac} \Gamma_c$,

$\Gamma^{bc} = \frac{1}{2} [\Gamma^b, \Gamma^c] = \frac{1}{2} (\Gamma^b \Gamma^c - \Gamma^c \Gamma^b)$,

$\Gamma^{n + 1} = -i^{k + t} \Gamma^1 ... \Gamma^n$

and similarly for the mathematical $\gamma$-matrices (in the first equation there is an implicit sum over $c$; this is an instance of the Einstein summation convention). These matrices satisfy by Lemma 6.3.6

$\{\Gamma^{n + 1}, \Gamma^a\} = 0$,

$[\Gamma^{n + 1}, \Gamma^{bc}] = 0$,

$\gamma^{bc} = -\Gamma^{bc}$.

In the following examples we use the Pauli matrices

$\sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $\sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$, $\sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$.

It is easy to check that they satisfy the identities

$\sigma^2 = I_2 \ \ j = 1, 2, 3$,

$\sigma_j \sigma_{j + 1} = -\sigma_{j + 1} \sigma_{j} = i \sigma_{j + 2} \ \ j = 1, 2, 3$,

where in the second equation ${j + 1}$ and $j + 2$ are taken $mod 3$.

$(\psi, \phi) = \psi^T C \phi$

Furthermore property 1. and 2. in Definition 6.7.1 are equivalent to

  1. $\gamma_a^T = \mu C \gamma_a C^{-1} \ \ for \ all \ a = 1, ..., s + t$.
  2. $C^T = \nu C$.

The first equation also holds with the physical Clifford matrices $\Gamma_a$ instead of the mathematical matrices $\gamma_a$.

There is an equivalent equation to the first one with physical Clifford matrices $\Gamma_a : 1 \cdot \Gamma^\dagger_a = -\delta A \Gamma_a A^{-1} \ \ for \ all \ a = 1, ..., s + t$.

Furthermore, property 1. and 2. in Definition 6.7.8 are equivalent to:

  1. $\gamma_a^{\dagger} = \delta A \gamma_a A^{-1} \ \ for \ all \ a = 1, ..., s + t$.
  2. $A^{\dagger} = A$.

Given a spin structure on a pseudo-Riemannian manifold and the spinor bundle $S$, we would like to have a covariant derivative on $S$ so that we can define field equations involving derivatives of spinors.

The Iconic Wall

corrected-wall

Tome

tome

Wrap Up

Why

The following three chapters discuss applications in physics: the Lagrangians and interactions in the Standard Model, spontaneous symmetry breaking, the Higgs mechanism of mass generation, and some more advanced and modern topics like neutrino masses and CP violation. Depending on the time, the interests and the prior knowledge of the reader, he or she can take a shortcut and immediately start at the chapters on connections, spinors or Lagrangians, and then go back if more detailed mathematical knowledge is required at some point. An interesting and perhaps underappreciated fact is that a substantial number of phenomena in particle physics can be understood by analysing representations of Lie groups and by rewriting or rearranging Lagrangians.

Symmetries of Lagrangians interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs boson), determined by the Lagrangian. For the symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In the following chapter we will study some associated concepts, like representations (which are used to define the actions of Lie groups on fields) and invariant matrices (which are important in the construction of the gauge invariant Yang-Mills Lagrangian). We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields.

The existence of positive definite Ad-invariant scalar products on the Lie algebra of compact Lie groups is very important in gauge theory, in particular, for the construction of the gauge-invariant Yang-Mills Lagrangian; see Sect. 7.3.1. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]). In a gauge-invariant Lagrangian this results in terms of order higher than two in the matter and gauge fields, which are interpreted as interactions between the corresponding particles. In non-abelian gauge theories, like quantum chromodynamics (QCD), there are also terms in the Lagrangian of order higher than two in the gauge fields themselves, coming from a quadratic term in the curvature that appears in the Yang-Mills Lagrangian.

In physics, the quadratic term $[A_\mu, A_\nu]$ in the expression for $F_{\mu\nu}$ (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field $A_\mu$. These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by $A_\mu$ and the particles described by the field $\phi$.

feynmandiagrams

Figure 5.2 shows the Feynman diagrams for the cubic and quartic terms which appear in the Klein-Gordon Lagrangian in Eq. (7.3), representing the interaction between a gauge field $A$ and a charged scalar field described locally by a map $\phi$ with values in $V$. Fig 5.2 Feynman diagrams for interaction between gauge field and charged scalar

Hermitian scalar products are particularly important, because we need them in Chap. 7 to define Lorentz invariant Lagrangians involving spinors.

$<\psi, \phi> \ = \ \overline{\psi} \phi$,

$\overline{\psi} = \psi^\dagger A$.

Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly the neutrinos) and, together with the Dirac operator the kinetic term and the interaction term; see Sect. 7.6.

Porta.jl

References

  1. Mark J.D. Hamilton, Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics, Springer Cham, DOI, published: 10 January 2018.

  2. Sir Roger Penrose, The Road to Reality, (2004).

  3. Roger Penrose, Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).

  4. Richard M. Murray and Zexiang Li, A Mathematical Introduction to Robotic Manipulation, 1st Edition, 1994, CRC Press, read, buy.

  5. Edward Witten, Physics and Geometry, (1987).

  6. The iconic Wall of Stony Brook University.

diff --git a/dev/reactionwheelunicycle.html b/dev/reactionwheelunicycle.html index e8f2218..ca9277c 100644 --- a/dev/reactionwheelunicycle.html +++ b/dev/reactionwheelunicycle.html @@ -1,2 +1,2 @@ -Reaction Wheel Unicycle · Porta.jl
GitHub

The Reaction Wheel Unicycle

$V_{cnt} = \begin{bmatrix} \dot{x} - r_w \dot{\theta} cos(\delta) \newline \dot{y} - r_w \dot{\theta} sin(\delta) \newline \dot{z} \end{bmatrix} = \begin{bmatrix} 0 \newline 0 \newline 0 \end{bmatrix}$

$\dot{x} = r_w \dot{\theta} cos(\delta)$

$\dot{y} = r_w \dot{\theta} sin(\delta)$

$\dot{z} = 0$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i}) - \frac{\partial L}{\partial q_i} = Q_i + \sum_{k=1}^n {\lambda}_k a_{ki}$

$i = 1, \ldots, m$

$L = T_{total} - P_{total}$

${}_{w2}^{cp}T = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\alpha) & -sin(\alpha) & 0 \newline 0 & sin(\alpha) & cos(\alpha) & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & r_w \newline 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\alpha) & -sin(\alpha) & -r_w sin(\alpha) \newline 0 & sin(\alpha) & cos(\alpha) & r_w cos(\alpha) \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_{cp}^{g}T = \begin{bmatrix} cos(\delta) & -sin(\delta) & 0 & x \newline sin(\delta) & cos(\delta) & 0 & y \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_{w2}^{g}T = {}_{cp}^{g}T \times {}_{w2}^{cp}T = \begin{bmatrix} cos(\delta) & -sin(\delta) cos(\alpha) & sin(\delta) sin(\alpha) & x + r_w sin(\delta) sin(\alpha) \newline sin(\delta) & cos(\delta) cos(\alpha) & -cos(\delta) sin(\alpha) & y - r_w cos(\delta) sin(\alpha) \newline 0 & sin(\alpha) & cos(\alpha) & r_w cos(\alpha) \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}^{w2}P_w = \begin{bmatrix} 0 \newline 0 \newline 0 \newline 1 \end{bmatrix}$

${}^gP_w = {}_{w2}^gT \times {}^{w2}P_w = \begin{bmatrix} x + r_w sin(\alpha) sin(\delta) \newline y - r_w sin(\alpha) cos(\delta) \newline r_w cos(\alpha) \newline 1 \end{bmatrix}$

${}_c^{w2}T = \begin{bmatrix} cos(\beta) & 0 & sin(\beta) & 0 \newline 0 & 1 & 0 & 0 \newline -sin(\beta) & 0 & cos(\beta) & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & l_c \newline 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} cos(\beta) & 0 & sin(\beta) & l_c sin(\beta) \newline 0 & 1 & 0 & 0 \newline -sin(\beta) & 0 & cos(\beta) & l_c cos(\beta) \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_c^gT = {}_{w2}^gT \times {}_c^{w2}T = \begin{bmatrix} {}_c^gt_{11} & -sin(\delta) cos(\alpha) & {}_c^gt_{13} & {}_c^gt_{14} \newline {}_c^gt_{21} & cos(\delta) cos(\alpha) & {}_c^gt_{23} & {}_c^gt_{24} \newline -cos(\alpha) sin(\beta) & sin(\alpha) & cos(\alpha) cos(\beta) & {}_c^gt_{34} \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_c^gt_{11} = cos(\beta) cos(\delta) - sin(\alpha) sin(beta) sin(\delta)$

${}_c^gt_{13} = sin(\beta) cos(\delta) + sin(\alpha) cos(\beta) sin(\delta)$

${}_c^gt_{14} = x + r_w sin(\delta) sin(\alpha) + l_c sin(\beta) cos(\delta) + l_c sin(\alpha) cos(\beta) sin(\delta)$

${}_c^gt_{21} = cos(\beta) sin(\delta) + sin(\alpha) sin(\beta) cos(\delta)$

${}_c^gt_{23} = sin(\beta) sin(\delta) - sin(\alpha) cos(\beta) cos(\delta)$

${}_c^gt_{24} = y - r_w cos(\delta) sin(\alpha) + l_c sin(\beta) sin(\delta) - l_c sin(\alpha) cos(\beta) cos(\delta)$

${}_c^gt_{34} = r_w cos(\alpha) + l_c cos(\alpha) cos(\beta)$

${}^cP_c = \begin{bmatrix} 0 \newline 0 \newline 0 \newline 1 \end{bmatrix}$

${}^gP_c = {}_c^gT \times {}^cP_c = \begin{bmatrix} {}^gp_{c1} \newline {}^gp_{c2} \newline {}^gp_{c3} \newline 1 \end{bmatrix}$

${}^gp_{c1} = x + r_w sin(\alpha) sin(\delta) + l_c cos(\beta) sin(\alpha) sin(\delta) + l_c sin(\beta) cos(\delta)$

${}^gp_{c2} = y - r_w sin(\alpha) cos(\delta) - l_c cos(\beta) sin(\alpha) cos(\delta) + l_c sin(\beta) sin(\delta)$

${}^gp_{c3} = r_w cos(\alpha) + l_c cos(\beta) cos(\alpha)$

${}_r^cT = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & l_{cr} \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\gamma) & -sin(\gamma) & 0 \newline 0 & sin(\gamma) & cos(\gamma) & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\gamma) & -sin(\gamma) & 0 \newline 0 & sin(\gamma) & cos(\gamma) & l_{cr} \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_r^gT = {}_c^gT \times {}_r^cT = \begin{bmatrix} {}_r^gt_{11} & {}_r^gt_{12} & {}_r^gt_{13} & {}_r^gt_{14} \newline {}_r^gt_{21} & {}_r^gt_{22} & {}_r^gt_{23} & {}_r^gt_{24} \newline -cos(\alpha) sin(\beta) & {}_r^gt_{32} & {}_r^gt_{33} & {}_r^gt_{34} \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_r^gt_{11} = cos(\beta) cos(\delta) - sin(\alpha) sin(\beta) sin(\delta)$

${}_r^gt_{12} = -sin(\delta) cos(\alpha) cos(\gamma) + cos(\delta) sin(\beta) sin(\gamma) + sin(\delta) sin(\alpha) cos(\beta) sin(\gamma)$

${}_r^gt_{13} = sin(\delta) cos(\alpha) sin(\gamma) + cos(\delta) sin(\beta) cos(\gamma) + sin(\delta) sin(\alpha) cos(\beta) cos(\gamma)$

${}_r^gt_{14} = 0 + l_{cr} (cos(\delta) sin(\beta) + sin(\delta) sin(\alpha) cos(\beta)) + l_c sin(\beta) cos(\delta) + l_c cos(\beta) sin(\delta) sin(\alpha) + x + r_w sin(\delta) sin(\alpha)$

${}_r^gt_{21} = cos(\beta) sin(\delta) + sin(\alpha) sin(\beta) cos(\delta)$

${}_r^gt_{22} = cos(\delta) cos(\alpha) cos(\gamma) + sin(\delta) sin(\beta) sin(\gamma) - cos(\delta) sin(\alpha) cos(\beta) sin(\gamma)$

${}_r^gt_{23} = -cos(\delta) cos(\alpha) sin(\gamma) + sin(\delta) sin(\beta) cos(\gamma) - cos(\delta) sin(\alpha) cos(\beta) cos(\gamma)$

${}_r^gt_{24} = l_{cr} (sin(\delta) sin(\beta) - cos(\delta) sin(\alpha) cos(\beta)) + l_c sin(\beta) sin(\delta) - l_c cos(\beta) cos(\delta) sin(\alpha) + y - r_w cos(\delta) sin(\alpha)$

${}_r^gt_{32} = sin(\alpha) cos(\gamma) + cos(\alpha) cos(\beta) sin(\gamma)$

${}_r^gt_{33} = -sin(\alpha) sin(\gamma) + cos(\alpha) cos(\beta) cos(\gamma)$

${}_r^gt_{34} = l_{cr} cos(\alpha) cos(\beta) + l_c cos(\beta) cos(\alpha) + r_w cos(\alpha)$

${}^rP_r = \begin{bmatrix} 0 \newline 0 \newline 0 \newline 1 \end{bmatrix}$

${}^gP_r = {}_r^gT \times {}^rP_r = \begin{bmatrix} {}^gp_{r1} \newline {}^gp_{r2} \newline {}^gp_{r3} \newline 1 \end{bmatrix}$

${}^gp_{r1} = x + r_w sin(\alpha) sin(\delta) + (l_c + l_{cr}) cos(\beta) sin(\alpha) sin(\delta) + (l_c + l_{cr}) sin(\beta) cos(\delta)$

${}^gp_{r2} = y - r_w sin(\alpha) cos(\delta) - (l_c + l_{cr}) cos(\beta) sin(\alpha) cos(\delta) + (l_c + l_{cr}) sin(\beta) sin(\delta)$

${}^gp_{r3} = r_w cos(\alpha) + (l_c + l_{cr}) cos(\beta) cos(\alpha)$

$V_w = \frac{dP_w}{dt}$

$V_c = \frac{dP_c}{dt}$

$V_r = \frac{dP_r}{dt}$

${\Omega}_w = \begin{bmatrix} 0 \newline \dot{\theta} \newline 0 \newline 0 \end{bmatrix} + \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_g^{w2}T \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} 0 \newline \dot{\theta} \newline 0 \newline 0 \end{bmatrix} + \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_{w2}^gT^{-1} \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\alpha} \newline \dot{\theta} + \dot{\delta} sin(\alpha) \newline \dot{\delta} cos(\alpha) \end{bmatrix}$

${\Omega}_c = \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_{w2}^cT \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_g^cT \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_c^{w2}T^{-1} \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_c^gT^{-1} \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\alpha} cos(\beta) - \dot{\delta} cos(\alpha) sin(\beta) \newline \dot{\beta} + \dot{\delta} sin(\alpha) \newline \dot{\alpha} sin(\beta) + \dot{\delta} cos(\alpha) cos(\beta) \newline 0 \end{bmatrix}$

${}_r^{w2}T = {}_{w2}^gT^{-1} \times {}_r^gT$

${\Omega}_r = \begin{bmatrix} \dot{\gamma} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_c^rT \times \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_{w2}^rT \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_g^rT \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\gamma} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_r^cT^{-1} \times \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_r^{w2}T^{-1} \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_r^gT^{-1} \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\gamma} + \dot{\alpha} cos(\beta) - \dot{\delta} cos(\alpha) sin(\beta) \newline {\omega}_{r2} \newline {\omega}_{r3} \newline 0 \end{bmatrix}$

${\omega}_{r2} = \dot{\beta} cos(\gamma) + \dot{\alpha} sin(\beta) sin(\gamma) + \dot{\delta} sin(\alpha) cos(\gamma) + \dot{\delta} cos(\alpha) cos(\beta) sin(\gamma)$

${\omega}_{r3} = -\dot{\beta} sin(\gamma) + \dot{\alpha} sin(\beta) cos(\gamma) - \dot{\delta} sin(\alpha) sin(\gamma) + \dot{\delta} cos(\alpha) cos(\beta) cos(\gamma)$

$T_w = \frac{1}{2} m_w V_w^T V_w + \frac{1}{2} {\Omega}_w^T I_w {\Omega}_w$

$P_w = m_w g P_w(3)$

$T_c = \frac{1}{2} m_c V_c^T V_c + \frac{1}{2} {\Omega}_c^T I_c {\Omega}_c$

$P_c = m_c g P_c(3)$

$T_r = \frac{1}{2} m_r V_r^T V_r + \frac{1}{2} {\Omega}_r^T I_r {\Omega}_r$

$P_r = m_r g P_r(3)$

$T_{total} = T_w + T_c + T_r$

$P_{total} = P_w + P_c + P_r$

$m = 7, \ n = 2$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{x}}) - \frac{\partial L}{\partial x} = {\lambda}_1$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{y}}) - \frac{\partial L}{\partial y} = {\lambda}_2$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\theta}}) - \frac{\partial L}{\partial \theta} = {\tau}_w - r_w cos(\delta) {\lambda}_1 - r_w sin(\delta) {\lambda}_2$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\beta}}) - \frac{\partial L}{\partial \beta} = -{\tau}_w$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\alpha}}) - \frac{\partial L}{\partial \alpha} = 0$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\gamma}}) - \frac{\partial L}{\partial \gamma} = {\tau}_r$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\delta}}) - \frac{\partial L}{\partial \delta} = 0$

Wheel dynamics:

$m_{11} \ddot{\beta} + m_{12} \ddot{\gamma} + m_{13} \ddot{\delta} + m_{14} \ddot{\theta} + c_{11} \dot{\beta}^2 + c_{12} \dot{\gamma}^2 + c_{13} \dot{\delta}^2 + c_{14} \dot{\alpha} \dot{\delta} + c_{15} \dot{\beta} \dot{\gamma} + c_{16} \dot{\beta} \dot{\delta} + c_{17} \dot{\gamma} \dot{\delta} = {\tau}_w$

Chassis longitudinal dynamics:

$m_{21} \ddot{\alpha} + m_{22} \ddot{\beta} + m_{23} \ddot{\delta} + m_{24} \ddot{\theta} + c_{21} \dot{\alpha}^2 + c_{22} \dot{\delta}^2 + c_{23} \dot{\alpha} \dot{\gamma} + c_{24} \dot{\alpha} \dot{\delta} + c_{25} \dot{\beta} \dot{\gamma} + c_{26} \dot{\gamma} \dot{\delta} + c_{27} \dot{\delta} \dot{\theta} + g_{21} = -{\tau}_w$

Chassis lateral dynamics:

$m_{31} \ddot{\alpha} + m_{32} \ddot{\beta} + m_{33} \ddot{\gamma} + m_{34} \ddot{\delta} + c_{31} \dot{\beta}^2 + c_{32} \dot{\gamma}^2 + c_{33} \dot{\delta}^2 + c_{34} \dot{\alpha} \dot{\beta} + c_{35} \dot{\alpha} \dot{\gamma} + c_{36} \dot{\beta} \dot{\gamma} + c_{37} \dot{\beta} \dot{\delta} + c_{38} \dot{\gamma} \dot{\delta} + c_{39} \dot{\delta} \dot{\theta} = 0$

Reaction wheel dynamics:

$m_{41} \ddot{\alpha} + m_{42} \ddot{\gamma} + m_{43} \ddot{\delta} + m_{44} \ddot{\theta} + c_{41} \dot{\alpha}^2 + c_{42} \dot{\beta}^2 + c_{43} \dot{\delta}^2 + c_{44} \dot{\alpha} \dot{\beta} + c_{45} \dot{\alpha} \dot{\delta} + c_{46} \dot{\beta} \dot{\delta} + c_{47} \dot{\delta} \dot{\theta} + g_{41} = {\tau}_r$

Turning dynamics:

$m_{51} \ddot{\alpha} + m_{52} \ddot{\beta} + m_{53} \ddot{\gamma} + m_{54} \ddot{\delta} + m_{55} \ddot{\theta} + c_{51} \dot{\alpha}^2 + c_{52} \dot{\beta}^2 + c_{53} \dot{\gamma}^2 + c_{54} \dot{\alpha} \dot{\beta} + c_{55} \dot{\alpha} \dot{\gamma} + c_{56} \dot{\alpha} \dot{\delta} + c_{57} \dot{\alpha} \dot{\theta} + c_{58} \dot{\beta} \dot{\gamma} + c_{59} \dot{\beta} \dot{\delta} + c_{510} \dot{\gamma} \dot{\delta} + c_{511} \dot{\delta} \dot{\theta} = 0$

$\frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} = r_{w} \cos\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \newline \frac{\mathrm{d} y\left( t \right)}{\mathrm{d}t} = r_{w} \sin\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \newline \frac{\mathrm{d} z\left( t \right)}{\mathrm{d}t} = 0 \newline I_{w} = \left[ \begin{array}{cccc} I_{w1} & 0 & 0 & 0 \newline 0 & I_{w2} & 0 & 0 \newline 0 & 0 & I_{w3} & 0 \newline 0 & 0 & 0 & 0 \newline \end{array} \right] \newline I_{c} = \left[ \begin{array}{cccc} I_{c1} & 0 & 0 & 0 \newline 0 & I_{c2} & 0 & 0 \newline 0 & 0 & I_{c3} & 0 \newline 0 & 0 & 0 & 0 \newline \end{array} \right] \newline I_{r} = \left[ \begin{array}{cccc} I_{r1} & 0 & 0 & 0 \newline 0 & I_{r2} & 0 & 0 \newline 0 & 0 & I_{r3} & 0 \newline 0 & 0 & 0 & 0 \newline \end{array} \right] \newline \mathrm{w2cpT}\left( t \right) = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \newline 0 & \cos\left( \alpha\left( t \right) \right) & - \sin\left( \alpha\left( t \right) \right) & - r_{w} \sin\left( \alpha\left( t \right) \right) \newline 0 & \sin\left( \alpha\left( t \right) \right) & \cos\left( \alpha\left( t \right) \right) & r_{w} \cos\left( \alpha\left( t \right) \right) \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{cpgT}\left( t \right) = \left[ \begin{array}{cccc} \cos\left( \delta\left( t \right) \right) & - \sin\left( \delta\left( t \right) \right) & 0 & x\left( t \right) \newline \sin\left( \delta\left( t \right) \right) & \cos\left( \delta\left( t \right) \right) & 0 & y\left( t \right) \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{w2gT}\left( t \right) = \mathrm{cpgT}\left( t \right) \mathrm{w2cpT}\left( t \right) \newline w2P_{w} = \left[ \begin{array}{c} 0 \newline 0 \newline 0 \newline 1 \newline \end{array} \right] \newline \mathrm{gP}_{w}\left( t \right) = \mathrm{w2gT}\left( t \right) w2P_{w} \newline \mathrm{cw2T}\left( t \right) = \left[ \begin{array}{cccc} \cos\left( \beta\left( t \right) \right) & 0 & \sin\left( \beta\left( t \right) \right) & l_{c} \sin\left( \beta\left( t \right) \right) \newline 0 & 1 & 0 & 0 \newline -\sin\left( \beta\left( t \right) \right) & 0 & \cos\left( \beta\left( t \right) \right) & l_{c} \cos\left( \beta\left( t \right) \right) \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{cgT}\left( t \right) = \mathrm{w2gT}\left( t \right) \mathrm{cw2T}\left( t \right) \newline cP_{c} = \left[ \begin{array}{c} 0 \newline 0 \newline 0 \newline 1 \newline \end{array} \right] \newline \mathrm{gP}_{c}\left( t \right) = \mathrm{cgT}\left( t \right) cP_{c} \newline \mathrm{rcT}\left( t \right) = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \newline 0 & \cos\left( \gamma\left( t \right) \right) & - \sin\left( \gamma\left( t \right) \right) & 0 \newline 0 & \sin\left( \gamma\left( t \right) \right) & \cos\left( \gamma\left( t \right) \right) & l_{cr} \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{rgT}\left( t \right) = \mathrm{cgT}\left( t \right) \mathrm{rcT}\left( t \right) \newline rP_{r} = \left[ \begin{array}{c} 0 \newline 0 \newline 0 \newline 1 \newline \end{array} \right] \newline \mathrm{gP}_{r}\left( t \right) = \mathrm{rgT}\left( t \right) rP_{r} \newline \mathrm{rw2T}\left( t \right) = \mathrm{inv}\left( \mathrm{w2gT}\left( t \right) \right) \mathrm{rgT}\left( t \right) \newline V_{w}\left( t \right) = \mathrm{broadcast}\left( D, \mathrm{gP}_{w}\left( t \right) \right) \newline V_{c}\left( t \right) = \mathrm{broadcast}\left( D, \mathrm{gP}_{c}\left( t \right) \right) \newline V_{r}\left( t \right) = \mathrm{broadcast}\left( D, \mathrm{gP}_{r}\left( t \right) \right) \newline \Omega_{w}\left( t \right) = \mathrm{broadcast}\left( +, \left[ \begin{array}{c} _{derivative}\left( \alpha\left( t \right), t, 1 \right) \newline _{derivative}\left( \theta\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline \end{array} \right], \mathrm{inv}\left( \mathrm{w2gT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline 0 \newline _{derivative}\left( \delta\left( t \right), t, 1 \right) \newline 0 \newline \end{array} \right] \right) \newline \Omega_{c}\left( t \right) = \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[ \begin{array}{c} 0 \newline _{derivative}\left( \beta\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline \end{array} \right], \mathrm{inv}\left( \mathrm{cw2T}\left( t \right) \right) \left[ \begin{array}{c} _{derivative}\left( \alpha\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline 0 \newline \end{array} \right] \right), \mathrm{inv}\left( \mathrm{cgT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline 0 \newline _{derivative}\left( \delta\left( t \right), t, 1 \right) \newline 0 \newline \end{array} \right] \right) \newline \Omega_{r}\left( t \right) = \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[ \begin{array}{c} _{derivative}\left( \gamma\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline 0 \newline \end{array} \right], \mathrm{inv}\left( \mathrm{rcT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline _{derivative}\left( \beta\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline \end{array} \right] \right), \mathrm{inv}\left( \mathrm{rw2T}\left( t \right) \right) \left[ \begin{array}{c} _{derivative}\left( \alpha\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline 0 \newline \end{array} \right] \right), \mathrm{inv}\left( \mathrm{rgT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline 0 \newline _{derivative}\left( \delta\left( t \right), t, 1 \right) \newline 0 \newline \end{array} \right] \right) \newline T_{w}\left( t \right) = \mathrm{adjoint}\left( V_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{w}\left( t \right), \mathrm{Ref}\left( 0.5 m_{w} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{w} \Omega_{w}\left( t \right), 0.5 \right)_{1} \newline P_{w}\left( t \right) = g \mathrm{gP}_{w}\left( t \right)_{3} m_{w} \newline T_{c}\left( t \right) = \mathrm{adjoint}\left( V_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{c}\left( t \right), \mathrm{Ref}\left( 0.5 m_{c} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{c} \Omega_{c}\left( t \right), 0.5 \right)_{1} \newline P_{c}\left( t \right) = g \mathrm{gP}_{c}\left( t \right)_{3} m_{c} \newline T_{r}\left( t \right) = \mathrm{adjoint}\left( V_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{r}\left( t \right), \mathrm{Ref}\left( 0.5 m_{r} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{r} \Omega_{r}\left( t \right), 0.5 \right)_{1} \newline P_{r}\left( t \right) = g \mathrm{gP}_{r}\left( t \right)_{3} m_{r} \newline T_{total}\left( t \right) = T_{r}\left( t \right) + T_{c}\left( t \right) + T_{w}\left( t \right) \newline P_{total}\left( t \right) = P_{w}\left( t \right) + P_{c}\left( t \right) + P_{r}\left( t \right) \newline L\left( t \right) = T_{total}\left( t \right) - P_{total}\left( t \right) \newline$

$L = 0.5 \left( \left( \frac{\frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t} \cos\left( \beta\left( t \right) \right)}{\sin^{2}\left( \beta\left( t \right) \right) + \cos^{2}\left( \beta\left( t \right) \right)} + \frac{\left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( - \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \beta\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right)} \right)^{2} I_{c1} + \left( \frac{\mathrm{d} \beta\left( t \right)}{\mathrm{d}t} + \frac{\left( - \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \left( - \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \beta\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right)} \right)^{2} I_{c2} + \left( \frac{\sin\left( \beta\left( t \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\sin^{2}\left( \beta\left( t \right) \right) + \cos^{2}\left( \beta\left( t \right) \right)} + \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \beta\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right)} \right)^{2} I_{c3} \right) + 0.5 \left( \left( \frac{\mathrm{d} \gamma\left( t \right)}{\mathrm{d}t} + \frac{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{ - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right)} + \frac{\left( \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \sin\left( \beta\left( t \right) \right) \right)} \right)^{2} I_{r1} + \left( \frac{\left( - \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) - \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{ - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right)} + \frac{\left( - \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \sin\left( \beta\left( t \right) \right) \right)} + \frac{\frac{\mathrm{d} \beta\left( t \right)}{\mathrm{d}t} \cos\left( \gamma\left( t \right) \right)}{\sin^{2}\left( \gamma\left( t \right) \right) + \cos^{2}\left( \gamma\left( t \right) \right)} \right)^{2} I_{r2} + \left( \frac{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{ - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right)} + \frac{\left( \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \sin\left( \beta\left( t \right) \right) \right)} + \frac{ - \frac{\mathrm{d} \beta\left( t \right)}{\mathrm{d}t} \sin\left( \gamma\left( t \right) \right)}{\sin^{2}\left( \gamma\left( t \right) \right) + \cos^{2}\left( \gamma\left( t \right) \right)} \right)^{2} I_{r3} \right) + 0.5 \left( \frac{\left( \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t} \right)^{2} \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right)^{2} I_{w3}}{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \right)^{2}} + \left( \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t} \right)^{2} I_{w1} + \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \right)^{2} I_{w2} \right) + 0.5 m_{c} \left( \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( x\left( t \right) + l_{c} \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + r_{w} \sin\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{c} \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} 1 \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( y\left( t \right) + l_{c} \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - r_{w} \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) - l_{c} \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} \right) + 0.5 m_{r} \left( \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( x\left( t \right) + l_{c} \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + r_{w} \sin\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{c} \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{cr} \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( y\left( t \right) + l_{c} \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - r_{w} \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) - l_{c} \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{cr} \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} 1 \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + l_{cr} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \right)^{2} \right) + 0.5 m_{w} \left( \left( \frac{\mathrm{d}}{\mathrm{d}t} r_{w} \cos\left( \alpha\left( t \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( y\left( t \right) - r_{w} \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( x\left( t \right) + r_{w} \sin\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} 1 \right)^{2} \right) - g m_{w} r_{w} \cos\left( \alpha\left( t \right) \right) - g \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) m_{c} - g \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + l_{cr} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) m_{r}$

$\left[ \begin{array}{c} _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline \end{array} \right] = \left[ \begin{array}{c} \lambda_1 \newline \lambda_2 \newline \tau_{w} - r_{w} \sin\left( \delta\left( t \right) \right) \lambda_2 - r_{w} \cos\left( \delta\left( t \right) \right) \lambda_1 \newline -\tau_{w} \newline 0 \newline \tau_{p} \newline 0 \newline \end{array} \right]$

+Reaction Wheel Unicycle · Porta.jl
GitHub

The Reaction Wheel Unicycle

$V_{cnt} = \begin{bmatrix} \dot{x} - r_w \dot{\theta} cos(\delta) \newline \dot{y} - r_w \dot{\theta} sin(\delta) \newline \dot{z} \end{bmatrix} = \begin{bmatrix} 0 \newline 0 \newline 0 \end{bmatrix}$

$\dot{x} = r_w \dot{\theta} cos(\delta)$

$\dot{y} = r_w \dot{\theta} sin(\delta)$

$\dot{z} = 0$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{q}_i}) - \frac{\partial L}{\partial q_i} = Q_i + \sum_{k=1}^n {\lambda}_k a_{ki}$

$i = 1, \ldots, m$

$L = T_{total} - P_{total}$

${}_{w2}^{cp}T = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\alpha) & -sin(\alpha) & 0 \newline 0 & sin(\alpha) & cos(\alpha) & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & r_w \newline 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\alpha) & -sin(\alpha) & -r_w sin(\alpha) \newline 0 & sin(\alpha) & cos(\alpha) & r_w cos(\alpha) \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_{cp}^{g}T = \begin{bmatrix} cos(\delta) & -sin(\delta) & 0 & x \newline sin(\delta) & cos(\delta) & 0 & y \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_{w2}^{g}T = {}_{cp}^{g}T \times {}_{w2}^{cp}T = \begin{bmatrix} cos(\delta) & -sin(\delta) cos(\alpha) & sin(\delta) sin(\alpha) & x + r_w sin(\delta) sin(\alpha) \newline sin(\delta) & cos(\delta) cos(\alpha) & -cos(\delta) sin(\alpha) & y - r_w cos(\delta) sin(\alpha) \newline 0 & sin(\alpha) & cos(\alpha) & r_w cos(\alpha) \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}^{w2}P_w = \begin{bmatrix} 0 \newline 0 \newline 0 \newline 1 \end{bmatrix}$

${}^gP_w = {}_{w2}^gT \times {}^{w2}P_w = \begin{bmatrix} x + r_w sin(\alpha) sin(\delta) \newline y - r_w sin(\alpha) cos(\delta) \newline r_w cos(\alpha) \newline 1 \end{bmatrix}$

${}_c^{w2}T = \begin{bmatrix} cos(\beta) & 0 & sin(\beta) & 0 \newline 0 & 1 & 0 & 0 \newline -sin(\beta) & 0 & cos(\beta) & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & l_c \newline 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} cos(\beta) & 0 & sin(\beta) & l_c sin(\beta) \newline 0 & 1 & 0 & 0 \newline -sin(\beta) & 0 & cos(\beta) & l_c cos(\beta) \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_c^gT = {}_{w2}^gT \times {}_c^{w2}T = \begin{bmatrix} {}_c^gt_{11} & -sin(\delta) cos(\alpha) & {}_c^gt_{13} & {}_c^gt_{14} \newline {}_c^gt_{21} & cos(\delta) cos(\alpha) & {}_c^gt_{23} & {}_c^gt_{24} \newline -cos(\alpha) sin(\beta) & sin(\alpha) & cos(\alpha) cos(\beta) & {}_c^gt_{34} \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_c^gt_{11} = cos(\beta) cos(\delta) - sin(\alpha) sin(beta) sin(\delta)$

${}_c^gt_{13} = sin(\beta) cos(\delta) + sin(\alpha) cos(\beta) sin(\delta)$

${}_c^gt_{14} = x + r_w sin(\delta) sin(\alpha) + l_c sin(\beta) cos(\delta) + l_c sin(\alpha) cos(\beta) sin(\delta)$

${}_c^gt_{21} = cos(\beta) sin(\delta) + sin(\alpha) sin(\beta) cos(\delta)$

${}_c^gt_{23} = sin(\beta) sin(\delta) - sin(\alpha) cos(\beta) cos(\delta)$

${}_c^gt_{24} = y - r_w cos(\delta) sin(\alpha) + l_c sin(\beta) sin(\delta) - l_c sin(\alpha) cos(\beta) cos(\delta)$

${}_c^gt_{34} = r_w cos(\alpha) + l_c cos(\alpha) cos(\beta)$

${}^cP_c = \begin{bmatrix} 0 \newline 0 \newline 0 \newline 1 \end{bmatrix}$

${}^gP_c = {}_c^gT \times {}^cP_c = \begin{bmatrix} {}^gp_{c1} \newline {}^gp_{c2} \newline {}^gp_{c3} \newline 1 \end{bmatrix}$

${}^gp_{c1} = x + r_w sin(\alpha) sin(\delta) + l_c cos(\beta) sin(\alpha) sin(\delta) + l_c sin(\beta) cos(\delta)$

${}^gp_{c2} = y - r_w sin(\alpha) cos(\delta) - l_c cos(\beta) sin(\alpha) cos(\delta) + l_c sin(\beta) sin(\delta)$

${}^gp_{c3} = r_w cos(\alpha) + l_c cos(\beta) cos(\alpha)$

${}_r^cT = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & l_{cr} \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\gamma) & -sin(\gamma) & 0 \newline 0 & sin(\gamma) & cos(\gamma) & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & cos(\gamma) & -sin(\gamma) & 0 \newline 0 & sin(\gamma) & cos(\gamma) & l_{cr} \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_r^gT = {}_c^gT \times {}_r^cT = \begin{bmatrix} {}_r^gt_{11} & {}_r^gt_{12} & {}_r^gt_{13} & {}_r^gt_{14} \newline {}_r^gt_{21} & {}_r^gt_{22} & {}_r^gt_{23} & {}_r^gt_{24} \newline -cos(\alpha) sin(\beta) & {}_r^gt_{32} & {}_r^gt_{33} & {}_r^gt_{34} \newline 0 & 0 & 0 & 1 \end{bmatrix}$

${}_r^gt_{11} = cos(\beta) cos(\delta) - sin(\alpha) sin(\beta) sin(\delta)$

${}_r^gt_{12} = -sin(\delta) cos(\alpha) cos(\gamma) + cos(\delta) sin(\beta) sin(\gamma) + sin(\delta) sin(\alpha) cos(\beta) sin(\gamma)$

${}_r^gt_{13} = sin(\delta) cos(\alpha) sin(\gamma) + cos(\delta) sin(\beta) cos(\gamma) + sin(\delta) sin(\alpha) cos(\beta) cos(\gamma)$

${}_r^gt_{14} = 0 + l_{cr} (cos(\delta) sin(\beta) + sin(\delta) sin(\alpha) cos(\beta)) + l_c sin(\beta) cos(\delta) + l_c cos(\beta) sin(\delta) sin(\alpha) + x + r_w sin(\delta) sin(\alpha)$

${}_r^gt_{21} = cos(\beta) sin(\delta) + sin(\alpha) sin(\beta) cos(\delta)$

${}_r^gt_{22} = cos(\delta) cos(\alpha) cos(\gamma) + sin(\delta) sin(\beta) sin(\gamma) - cos(\delta) sin(\alpha) cos(\beta) sin(\gamma)$

${}_r^gt_{23} = -cos(\delta) cos(\alpha) sin(\gamma) + sin(\delta) sin(\beta) cos(\gamma) - cos(\delta) sin(\alpha) cos(\beta) cos(\gamma)$

${}_r^gt_{24} = l_{cr} (sin(\delta) sin(\beta) - cos(\delta) sin(\alpha) cos(\beta)) + l_c sin(\beta) sin(\delta) - l_c cos(\beta) cos(\delta) sin(\alpha) + y - r_w cos(\delta) sin(\alpha)$

${}_r^gt_{32} = sin(\alpha) cos(\gamma) + cos(\alpha) cos(\beta) sin(\gamma)$

${}_r^gt_{33} = -sin(\alpha) sin(\gamma) + cos(\alpha) cos(\beta) cos(\gamma)$

${}_r^gt_{34} = l_{cr} cos(\alpha) cos(\beta) + l_c cos(\beta) cos(\alpha) + r_w cos(\alpha)$

${}^rP_r = \begin{bmatrix} 0 \newline 0 \newline 0 \newline 1 \end{bmatrix}$

${}^gP_r = {}_r^gT \times {}^rP_r = \begin{bmatrix} {}^gp_{r1} \newline {}^gp_{r2} \newline {}^gp_{r3} \newline 1 \end{bmatrix}$

${}^gp_{r1} = x + r_w sin(\alpha) sin(\delta) + (l_c + l_{cr}) cos(\beta) sin(\alpha) sin(\delta) + (l_c + l_{cr}) sin(\beta) cos(\delta)$

${}^gp_{r2} = y - r_w sin(\alpha) cos(\delta) - (l_c + l_{cr}) cos(\beta) sin(\alpha) cos(\delta) + (l_c + l_{cr}) sin(\beta) sin(\delta)$

${}^gp_{r3} = r_w cos(\alpha) + (l_c + l_{cr}) cos(\beta) cos(\alpha)$

$V_w = \frac{dP_w}{dt}$

$V_c = \frac{dP_c}{dt}$

$V_r = \frac{dP_r}{dt}$

${\Omega}_w = \begin{bmatrix} 0 \newline \dot{\theta} \newline 0 \newline 0 \end{bmatrix} + \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_g^{w2}T \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} 0 \newline \dot{\theta} \newline 0 \newline 0 \end{bmatrix} + \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_{w2}^gT^{-1} \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\alpha} \newline \dot{\theta} + \dot{\delta} sin(\alpha) \newline \dot{\delta} cos(\alpha) \end{bmatrix}$

${\Omega}_c = \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_{w2}^cT \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_g^cT \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_c^{w2}T^{-1} \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_c^gT^{-1} \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\alpha} cos(\beta) - \dot{\delta} cos(\alpha) sin(\beta) \newline \dot{\beta} + \dot{\delta} sin(\alpha) \newline \dot{\alpha} sin(\beta) + \dot{\delta} cos(\alpha) cos(\beta) \newline 0 \end{bmatrix}$

${}_r^{w2}T = {}_{w2}^gT^{-1} \times {}_r^gT$

${\Omega}_r = \begin{bmatrix} \dot{\gamma} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_c^rT \times \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_{w2}^rT \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_g^rT \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\gamma} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_r^cT^{-1} \times \begin{bmatrix} 0 \newline \dot{\beta} \newline 0 \newline 0 \end{bmatrix} + {}_r^{w2}T^{-1} \times \begin{bmatrix} \dot{\alpha} \newline 0 \newline 0 \newline 0 \end{bmatrix} + {}_r^gT^{-1} \times \begin{bmatrix} 0 \newline 0 \newline \dot{\delta} \newline 0 \end{bmatrix} = \begin{bmatrix} \dot{\gamma} + \dot{\alpha} cos(\beta) - \dot{\delta} cos(\alpha) sin(\beta) \newline {\omega}_{r2} \newline {\omega}_{r3} \newline 0 \end{bmatrix}$

${\omega}_{r2} = \dot{\beta} cos(\gamma) + \dot{\alpha} sin(\beta) sin(\gamma) + \dot{\delta} sin(\alpha) cos(\gamma) + \dot{\delta} cos(\alpha) cos(\beta) sin(\gamma)$

${\omega}_{r3} = -\dot{\beta} sin(\gamma) + \dot{\alpha} sin(\beta) cos(\gamma) - \dot{\delta} sin(\alpha) sin(\gamma) + \dot{\delta} cos(\alpha) cos(\beta) cos(\gamma)$

$T_w = \frac{1}{2} m_w V_w^T V_w + \frac{1}{2} {\Omega}_w^T I_w {\Omega}_w$

$P_w = m_w g P_w(3)$

$T_c = \frac{1}{2} m_c V_c^T V_c + \frac{1}{2} {\Omega}_c^T I_c {\Omega}_c$

$P_c = m_c g P_c(3)$

$T_r = \frac{1}{2} m_r V_r^T V_r + \frac{1}{2} {\Omega}_r^T I_r {\Omega}_r$

$P_r = m_r g P_r(3)$

$T_{total} = T_w + T_c + T_r$

$P_{total} = P_w + P_c + P_r$

$m = 7, \ n = 2$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{x}}) - \frac{\partial L}{\partial x} = {\lambda}_1$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{y}}) - \frac{\partial L}{\partial y} = {\lambda}_2$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\theta}}) - \frac{\partial L}{\partial \theta} = {\tau}_w - r_w cos(\delta) {\lambda}_1 - r_w sin(\delta) {\lambda}_2$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\beta}}) - \frac{\partial L}{\partial \beta} = -{\tau}_w$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\alpha}}) - \frac{\partial L}{\partial \alpha} = 0$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\gamma}}) - \frac{\partial L}{\partial \gamma} = {\tau}_r$

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{\delta}}) - \frac{\partial L}{\partial \delta} = 0$

Wheel dynamics:

$m_{11} \ddot{\beta} + m_{12} \ddot{\gamma} + m_{13} \ddot{\delta} + m_{14} \ddot{\theta} + c_{11} \dot{\beta}^2 + c_{12} \dot{\gamma}^2 + c_{13} \dot{\delta}^2 + c_{14} \dot{\alpha} \dot{\delta} + c_{15} \dot{\beta} \dot{\gamma} + c_{16} \dot{\beta} \dot{\delta} + c_{17} \dot{\gamma} \dot{\delta} = {\tau}_w$

Chassis longitudinal dynamics:

$m_{21} \ddot{\alpha} + m_{22} \ddot{\beta} + m_{23} \ddot{\delta} + m_{24} \ddot{\theta} + c_{21} \dot{\alpha}^2 + c_{22} \dot{\delta}^2 + c_{23} \dot{\alpha} \dot{\gamma} + c_{24} \dot{\alpha} \dot{\delta} + c_{25} \dot{\beta} \dot{\gamma} + c_{26} \dot{\gamma} \dot{\delta} + c_{27} \dot{\delta} \dot{\theta} + g_{21} = -{\tau}_w$

Chassis lateral dynamics:

$m_{31} \ddot{\alpha} + m_{32} \ddot{\beta} + m_{33} \ddot{\gamma} + m_{34} \ddot{\delta} + c_{31} \dot{\beta}^2 + c_{32} \dot{\gamma}^2 + c_{33} \dot{\delta}^2 + c_{34} \dot{\alpha} \dot{\beta} + c_{35} \dot{\alpha} \dot{\gamma} + c_{36} \dot{\beta} \dot{\gamma} + c_{37} \dot{\beta} \dot{\delta} + c_{38} \dot{\gamma} \dot{\delta} + c_{39} \dot{\delta} \dot{\theta} = 0$

Reaction wheel dynamics:

$m_{41} \ddot{\alpha} + m_{42} \ddot{\gamma} + m_{43} \ddot{\delta} + m_{44} \ddot{\theta} + c_{41} \dot{\alpha}^2 + c_{42} \dot{\beta}^2 + c_{43} \dot{\delta}^2 + c_{44} \dot{\alpha} \dot{\beta} + c_{45} \dot{\alpha} \dot{\delta} + c_{46} \dot{\beta} \dot{\delta} + c_{47} \dot{\delta} \dot{\theta} + g_{41} = {\tau}_r$

Turning dynamics:

$m_{51} \ddot{\alpha} + m_{52} \ddot{\beta} + m_{53} \ddot{\gamma} + m_{54} \ddot{\delta} + m_{55} \ddot{\theta} + c_{51} \dot{\alpha}^2 + c_{52} \dot{\beta}^2 + c_{53} \dot{\gamma}^2 + c_{54} \dot{\alpha} \dot{\beta} + c_{55} \dot{\alpha} \dot{\gamma} + c_{56} \dot{\alpha} \dot{\delta} + c_{57} \dot{\alpha} \dot{\theta} + c_{58} \dot{\beta} \dot{\gamma} + c_{59} \dot{\beta} \dot{\delta} + c_{510} \dot{\gamma} \dot{\delta} + c_{511} \dot{\delta} \dot{\theta} = 0$

$\frac{\mathrm{d} x\left( t \right)}{\mathrm{d}t} = r_{w} \cos\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \newline \frac{\mathrm{d} y\left( t \right)}{\mathrm{d}t} = r_{w} \sin\left( \delta\left( t \right) \right) \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \newline \frac{\mathrm{d} z\left( t \right)}{\mathrm{d}t} = 0 \newline I_{w} = \left[ \begin{array}{cccc} I_{w1} & 0 & 0 & 0 \newline 0 & I_{w2} & 0 & 0 \newline 0 & 0 & I_{w3} & 0 \newline 0 & 0 & 0 & 0 \newline \end{array} \right] \newline I_{c} = \left[ \begin{array}{cccc} I_{c1} & 0 & 0 & 0 \newline 0 & I_{c2} & 0 & 0 \newline 0 & 0 & I_{c3} & 0 \newline 0 & 0 & 0 & 0 \newline \end{array} \right] \newline I_{r} = \left[ \begin{array}{cccc} I_{r1} & 0 & 0 & 0 \newline 0 & I_{r2} & 0 & 0 \newline 0 & 0 & I_{r3} & 0 \newline 0 & 0 & 0 & 0 \newline \end{array} \right] \newline \mathrm{w2cpT}\left( t \right) = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \newline 0 & \cos\left( \alpha\left( t \right) \right) & - \sin\left( \alpha\left( t \right) \right) & - r_{w} \sin\left( \alpha\left( t \right) \right) \newline 0 & \sin\left( \alpha\left( t \right) \right) & \cos\left( \alpha\left( t \right) \right) & r_{w} \cos\left( \alpha\left( t \right) \right) \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{cpgT}\left( t \right) = \left[ \begin{array}{cccc} \cos\left( \delta\left( t \right) \right) & - \sin\left( \delta\left( t \right) \right) & 0 & x\left( t \right) \newline \sin\left( \delta\left( t \right) \right) & \cos\left( \delta\left( t \right) \right) & 0 & y\left( t \right) \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{w2gT}\left( t \right) = \mathrm{cpgT}\left( t \right) \mathrm{w2cpT}\left( t \right) \newline w2P_{w} = \left[ \begin{array}{c} 0 \newline 0 \newline 0 \newline 1 \newline \end{array} \right] \newline \mathrm{gP}_{w}\left( t \right) = \mathrm{w2gT}\left( t \right) w2P_{w} \newline \mathrm{cw2T}\left( t \right) = \left[ \begin{array}{cccc} \cos\left( \beta\left( t \right) \right) & 0 & \sin\left( \beta\left( t \right) \right) & l_{c} \sin\left( \beta\left( t \right) \right) \newline 0 & 1 & 0 & 0 \newline -\sin\left( \beta\left( t \right) \right) & 0 & \cos\left( \beta\left( t \right) \right) & l_{c} \cos\left( \beta\left( t \right) \right) \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{cgT}\left( t \right) = \mathrm{w2gT}\left( t \right) \mathrm{cw2T}\left( t \right) \newline cP_{c} = \left[ \begin{array}{c} 0 \newline 0 \newline 0 \newline 1 \newline \end{array} \right] \newline \mathrm{gP}_{c}\left( t \right) = \mathrm{cgT}\left( t \right) cP_{c} \newline \mathrm{rcT}\left( t \right) = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \newline 0 & \cos\left( \gamma\left( t \right) \right) & - \sin\left( \gamma\left( t \right) \right) & 0 \newline 0 & \sin\left( \gamma\left( t \right) \right) & \cos\left( \gamma\left( t \right) \right) & l_{cr} \newline 0 & 0 & 0 & 1 \newline \end{array} \right] \newline \mathrm{rgT}\left( t \right) = \mathrm{cgT}\left( t \right) \mathrm{rcT}\left( t \right) \newline rP_{r} = \left[ \begin{array}{c} 0 \newline 0 \newline 0 \newline 1 \newline \end{array} \right] \newline \mathrm{gP}_{r}\left( t \right) = \mathrm{rgT}\left( t \right) rP_{r} \newline \mathrm{rw2T}\left( t \right) = \mathrm{inv}\left( \mathrm{w2gT}\left( t \right) \right) \mathrm{rgT}\left( t \right) \newline V_{w}\left( t \right) = \mathrm{broadcast}\left( D, \mathrm{gP}_{w}\left( t \right) \right) \newline V_{c}\left( t \right) = \mathrm{broadcast}\left( D, \mathrm{gP}_{c}\left( t \right) \right) \newline V_{r}\left( t \right) = \mathrm{broadcast}\left( D, \mathrm{gP}_{r}\left( t \right) \right) \newline \Omega_{w}\left( t \right) = \mathrm{broadcast}\left( +, \left[ \begin{array}{c} _{derivative}\left( \alpha\left( t \right), t, 1 \right) \newline _{derivative}\left( \theta\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline \end{array} \right], \mathrm{inv}\left( \mathrm{w2gT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline 0 \newline _{derivative}\left( \delta\left( t \right), t, 1 \right) \newline 0 \newline \end{array} \right] \right) \newline \Omega_{c}\left( t \right) = \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[ \begin{array}{c} 0 \newline _{derivative}\left( \beta\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline \end{array} \right], \mathrm{inv}\left( \mathrm{cw2T}\left( t \right) \right) \left[ \begin{array}{c} _{derivative}\left( \alpha\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline 0 \newline \end{array} \right] \right), \mathrm{inv}\left( \mathrm{cgT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline 0 \newline _{derivative}\left( \delta\left( t \right), t, 1 \right) \newline 0 \newline \end{array} \right] \right) \newline \Omega_{r}\left( t \right) = \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \mathrm{broadcast}\left( +, \left[ \begin{array}{c} _{derivative}\left( \gamma\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline 0 \newline \end{array} \right], \mathrm{inv}\left( \mathrm{rcT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline _{derivative}\left( \beta\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline \end{array} \right] \right), \mathrm{inv}\left( \mathrm{rw2T}\left( t \right) \right) \left[ \begin{array}{c} _{derivative}\left( \alpha\left( t \right), t, 1 \right) \newline 0 \newline 0 \newline 0 \newline \end{array} \right] \right), \mathrm{inv}\left( \mathrm{rgT}\left( t \right) \right) \left[ \begin{array}{c} 0 \newline 0 \newline _{derivative}\left( \delta\left( t \right), t, 1 \right) \newline 0 \newline \end{array} \right] \right) \newline T_{w}\left( t \right) = \mathrm{adjoint}\left( V_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{w}\left( t \right), \mathrm{Ref}\left( 0.5 m_{w} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{w}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{w} \Omega_{w}\left( t \right), 0.5 \right)_{1} \newline P_{w}\left( t \right) = g \mathrm{gP}_{w}\left( t \right)_{3} m_{w} \newline T_{c}\left( t \right) = \mathrm{adjoint}\left( V_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{c}\left( t \right), \mathrm{Ref}\left( 0.5 m_{c} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{c}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{c} \Omega_{c}\left( t \right), 0.5 \right)_{1} \newline P_{c}\left( t \right) = g \mathrm{gP}_{c}\left( t \right)_{3} m_{c} \newline T_{r}\left( t \right) = \mathrm{adjoint}\left( V_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, V_{r}\left( t \right), \mathrm{Ref}\left( 0.5 m_{r} \right) \right)_{1} + \mathrm{adjoint}\left( \Omega_{r}\left( t \right) \right) \mathrm{broadcast}\left( *, I_{r} \Omega_{r}\left( t \right), 0.5 \right)_{1} \newline P_{r}\left( t \right) = g \mathrm{gP}_{r}\left( t \right)_{3} m_{r} \newline T_{total}\left( t \right) = T_{r}\left( t \right) + T_{c}\left( t \right) + T_{w}\left( t \right) \newline P_{total}\left( t \right) = P_{w}\left( t \right) + P_{c}\left( t \right) + P_{r}\left( t \right) \newline L\left( t \right) = T_{total}\left( t \right) - P_{total}\left( t \right) \newline$

$L = 0.5 \left( \left( \frac{\frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t} \cos\left( \beta\left( t \right) \right)}{\sin^{2}\left( \beta\left( t \right) \right) + \cos^{2}\left( \beta\left( t \right) \right)} + \frac{\left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( - \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \beta\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right)} \right)^{2} I_{c1} + \left( \frac{\mathrm{d} \beta\left( t \right)}{\mathrm{d}t} + \frac{\left( - \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \left( - \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \beta\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right)} \right)^{2} I_{c2} + \left( \frac{\sin\left( \beta\left( t \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\sin^{2}\left( \beta\left( t \right) \right) + \cos^{2}\left( \beta\left( t \right) \right)} + \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \beta\left( t \right) \right) \right) + \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) + \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right) \right)} \right)^{2} I_{c3} \right) + 0.5 \left( \left( \frac{\mathrm{d} \gamma\left( t \right)}{\mathrm{d}t} + \frac{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{ - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right)} + \frac{\left( \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \sin\left( \beta\left( t \right) \right) \right)} \right)^{2} I_{r1} + \left( \frac{\left( - \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) - \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{ - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right)} + \frac{\left( - \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \sin\left( \beta\left( t \right) \right) \right)} + \frac{\frac{\mathrm{d} \beta\left( t \right)}{\mathrm{d}t} \cos\left( \gamma\left( t \right) \right)}{\sin^{2}\left( \gamma\left( t \right) \right) + \cos^{2}\left( \gamma\left( t \right) \right)} \right)^{2} I_{r2} + \left( \frac{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t}}{\left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{ - \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \left( \frac{ - \sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\cos\left( \alpha\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) + \left( \frac{\sin\left( \delta\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \cos\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \left( \frac{\sin\left( \delta\left( t \right) \right) \left( - \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{ - \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right) \right) \left( \frac{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin\left( \delta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right)}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} \right)} + \frac{\left( \left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\left( \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) - \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \sin\left( \gamma\left( t \right) \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \sin\left( \beta\left( t \right) \right) \left( \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \cos\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( - \sin\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) + \left( \sin\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) + \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \cos\left( \gamma\left( t \right) \right) \right) \left( \left( \cos\left( \gamma\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos\left( \alpha\left( t \right) \right) \sin\left( \gamma\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \left( \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) - \cos\left( \alpha\left( t \right) \right) \left( - \cos\left( \alpha\left( t \right) \right) \cos\left( \gamma\left( t \right) \right) \cos\left( \delta\left( t \right) \right) - \sin\left( \gamma\left( t \right) \right) \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \sin\left( \beta\left( t \right) \right) \right)} + \frac{ - \frac{\mathrm{d} \beta\left( t \right)}{\mathrm{d}t} \sin\left( \gamma\left( t \right) \right)}{\sin^{2}\left( \gamma\left( t \right) \right) + \cos^{2}\left( \gamma\left( t \right) \right)} \right)^{2} I_{r3} \right) + 0.5 \left( \frac{\left( \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t} \right)^{2} \left( \sin^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \cos\left( \alpha\left( t \right) \right) \right)^{2} I_{w3}}{\left( \cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right) \right)^{2}} + \left( \frac{\mathrm{d} \alpha\left( t \right)}{\mathrm{d}t} \right)^{2} I_{w1} + \left( \frac{\left( \sin^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + \cos^{2}\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \frac{\mathrm{d} \delta\left( t \right)}{\mathrm{d}t}}{\cos^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \sin^{2}\left( \delta\left( t \right) \right) + \left( \cos^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin^{2}\left( \alpha\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \right) \cos\left( \delta\left( t \right) \right)} + \frac{\mathrm{d} \theta\left( t \right)}{\mathrm{d}t} \right)^{2} I_{w2} \right) + 0.5 m_{c} \left( \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( x\left( t \right) + l_{c} \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + r_{w} \sin\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{c} \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} 1 \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( y\left( t \right) + l_{c} \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - r_{w} \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) - l_{c} \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} \right) + 0.5 m_{r} \left( \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( x\left( t \right) + l_{c} \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + r_{w} \sin\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{c} \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{cr} \left( \sin\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) + \sin\left( \delta\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( y\left( t \right) + l_{c} \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - r_{w} \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) - l_{c} \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) + l_{cr} \left( \sin\left( \delta\left( t \right) \right) \sin\left( \beta\left( t \right) \right) - \cos\left( \beta\left( t \right) \right) \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} 1 \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + l_{cr} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) \right)^{2} \right) + 0.5 m_{w} \left( \left( \frac{\mathrm{d}}{\mathrm{d}t} r_{w} \cos\left( \alpha\left( t \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( y\left( t \right) - r_{w} \cos\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} \left( x\left( t \right) + r_{w} \sin\left( \delta\left( t \right) \right) \sin\left( \alpha\left( t \right) \right) \right) \right)^{2} + \left( \frac{\mathrm{d}}{\mathrm{d}t} 1 \right)^{2} \right) - g m_{w} r_{w} \cos\left( \alpha\left( t \right) \right) - g \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) m_{c} - g \left( r_{w} \cos\left( \alpha\left( t \right) \right) + l_{c} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) + l_{cr} \cos\left( \alpha\left( t \right) \right) \cos\left( \beta\left( t \right) \right) \right) m_{r}$

$\left[ \begin{array}{c} _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline _{derivative}\left( 0, t, 1 \right) \newline \end{array} \right] = \left[ \begin{array}{c} \lambda_1 \newline \lambda_2 \newline \tau_{w} - r_{w} \sin\left( \delta\left( t \right) \right) \lambda_2 - r_{w} \cos\left( \delta\left( t \right) \right) \lambda_1 \newline -\tau_{w} \newline 0 \newline \tau_{p} \newline 0 \newline \end{array} \right]$

diff --git a/dev/search_index.js b/dev/search_index.js index 9f65a21..b50eca2 100644 --- a/dev/search_index.js +++ b/dev/search_index.js @@ -1,3 +1,3 @@ var documenterSearchIndex = {"docs": -[{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Description = \"How the Hopf fibration works.\"","category":"page"},{"location":"hopffibration.html#The-Hopf-Fibration","page":"Hopf Fibration","title":"The Hopf Fibration","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The Hopf fibration is a fiber bundle with a two-dimensional sphere as the base space and circles as the fiber space. It is the geometrical shape that relates Einstein's spacetime to quantum fields. In this model, we visualize the Hopf fibration by first computing its points via a bundle atlas and then rendering the points in 3D space via stereographic projection. The projection step is necessary because the Hopf fibration is embedded in a four-space. Yet, it has only three degrees of freedom as a three-dimensional shape. The idea that makes this model more special and interesting than a typical visualization is the idea of Planet Hopf, due to Dror Bar-Natan (2010). The basic idea is that since the Hopf map takes the three-dimensional sphere into the two-dimensional sphere, we can pull the skin of the globe back to the three-sphere and visualize it.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Into the bargain, the Earth rotates about its axis every 24 hours. That spinning transformation of the Earth, together with the non-trivial product space of the Hopf bundle, can be encoded naturally into a monolithic visualization. It also makes sense to visualize differential operators in the Minkowski space-time as vectors in a cross-section of the Hopf bundle and then study the properties of spin-transformations. The choice of a gauge transformation (or trivialization) along with Lorentz transformations of Minkowski spacetime should not have any effect on physical laws. It is therefore a great model to understand these transformations and walk the road to reality. The following explains how the source code for generating animations of the Hopf fibration works (alternative views of Planet Hopf). We follow the beginning of chapter 4 of Mark J.D. Hamilton (2018) for a formal definition of the Hopf fibration as a fiber bundle. The book Mathematical Gauge Theory explains the Standard Model to students of both mathematics and physics, covers both the specific gauge theory of the Standard Model and generalizations, and is highly accessible and self-contained. Then, the definitions are going to be used to explain the source code in terms of computational methods and types.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image1) (Image: board2) (Image: image2)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"First, let E and M be smooth manifolds. Then, pi E to M is a surjective and differentiable map between smooth manifolds. Meaning, every element in M has some corresponding element in E via the map pi. Now, let x in M be a point. A fiber of pi over point x is called E_x and defined as a non-empty subset of E as follows: E_x = pi^-1(x) = pi^-1(x) subset E. The singleton of x is taken to the manifold E by the inverse of the map pi. However, to have a set of more than one point let U be a subset of M, U subset M. Then, we have E_U = pi^-1(U) subset E. In this case, E_U is the part of E above the subset U.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image3) (Image: board3) (Image: image4)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, define a global section of the map pi like this: s M to E. Considering the definition of pi E to M, the definition of the global section implies that the composition of pi and s is the identity map pi o s = Id_M over M. A section such as s can be a local one if we take a subset of M in the domain, U subset M. Then, a local section is defined as s U to E. In a similar way the definition of the local section implies that its composition with pi is the idenity map over the subset: pi o s = Id_U. For all points x in subset U, the section s(x) is in the fiber E_x of pi above x, if and only if s is a local section of pi. In this pointwise case, the map pi is restricted to subset U. In other words pi E to U, where U subset M.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image5) (Image: board4) (Image: image6)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In general, for two points x not = y in M that are not equal, the fibers E_x and E_y of pi over x and y may not be embedded submanifolds of E, or even be diffeomorphic. That means, there may not be a differentiable and invertible map that takes fiber E_x into fiber E_y, and the tangent spaces of E_x and E_y over points x and y may not be naturally linear subspaces of the tangent space of E. But, it is different in the special instance where manifold E = M times F is the product of M and the general fiber F and pi as a map is the projection onto the first factor pi M times F to M. If that is the case, then fibers E_x E_y in F of pi over the two distinct points x not = y in M are embedded submanifolds of E and diffeomorphic. To explain it more clearly, given that condition, there exists an invertible and smooth map taking one fiber to the other, and the tangent spaces of the fibers are directly summed with their respective dual subspaces at points in the fibers to span the whole tangent space of manifold E at points of pi over x and y. Therefore, fiber bundles are the generalization of products E = M times F as twisted products.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image7) (Image: board5) (Image: image8)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Before we define the Hopf action, first describe a scalar multiplication rule between vectors and numbers. Let R denote real numbers, Complex complex numbers, and mathbbH quaternionic numbers. On top of that, take a subset of these sets of numbers such that zero is not allowed to be in them, and denote the subsets as R^*, Complex^*, and mathbbH^* respectively. Now, define the linear right action by scalar multiplication for mathbbK = mathbbR mathbbC mathbbH as the following: mathbbK^n+1setminus0 times mathbbK^* to mathbbK^n+1setminus0. For example, 5 in mathbbR^* is a non-zero scalar number, whereas 1 0 0^T in mathbbR^3setminus0 is a non-zero vector quantity. Per our definition, 5 acts on 1 0 0^T on the right and yields 5 0 0 in mathbbR^3setminus0 as another vector. This rule works the same for fields mathbbK even when the vectorial numbers are represented by matrices.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image9) (Image: board6) (Image: image10)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The linear right action by multiplication is called a free action, because for x in mathbbK^n+1setminus0 and y in mathbbk^* the multiplication x times y yields x if and only if y = Id, as the identity element. For example, if we let x = 0 1 0^T y = 1, then the result of the scalar multiplication is 0 1 0^T times 1 = 0 1 0^T.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image11) (Image: board7) (Image: image12)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In addition, we define the unit n-sphere, for the Hopf action works on spheres. So, the unit sphere of dimension n is defined as: S^n(w_1 w_2 w_n+1) in mathbbR^n+1 sum_substack1 leq i leq n+1w_i^2 = 1. As an example, the unit circle S^1 in mathbbC is a one-dimensional sphere with n = 1, and w_1^2 + w_2^2 = 1, where w_1 and w_2 are the horizontal and vertical axes in the complex plane, respectively.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image13) (Image: board8) (Image: image14)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Now, Hopf actions are defined as free actions:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"S^n times S^0 to S^n \nS^2n+1 times S^1 to S^2n+1 \nS^4n+3 times S^3 to S^4n+3 ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"given by (x lambda) mapsto xlambda.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image15) (Image: board9) (Image: image16)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"An example of a Hopf action is the multiplication of the three-sphere S^3 cong SU(2) subset mathbbC^2 on the right by the unit circle S^1 cong U(1) subset mathbbC. Define the Hopf action as the map Phi S^3 times S^1 to S^3 given by (v w lambda) mapsto (v w) sdot lambda = (vlambda wlambda), for all points in the unit 3-sphere (v w) in S^3 and the unit 1-sphere lambda in S^1. What's more, the Hopf action has two properties:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(v w) sdot (lambda sdot mu) = ((v w) sdot lambda) sdot mu\n(v w) sdot 1 = (v w)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"forall (v w) in S^3 lambda mu in S^1.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image17) (Image: board10) (Image: image18)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The next idea is about the orbit of a point in the 3-sphere S^3 under the Hopf action. The orbit map is defined as phi S^1 to S^3 given by lambda mapsto (v_0 w_0) sdot lambda, forall (v_0 w_0) in S^3. The orbit map phi is injective and free, meaning that a point in S^3 can not have many points in S^1 and also there exists an identity element such that the action stabalizes a point in S^3 such as (v_0 w_0). Furthermore, the Hopf action Phi S^1 to Diff(S^3) is a homomorphism. It preserves S^3. The Hopf action being a free action implies that the orbit of every point (v_0 w_0) in S^3 is an embedded circle S^1.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image19) (Image: board11) (Image: image20)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Back to the topic of fiber bundles, we recall that the part of manifold E over subset U equals: E_U = pi^-1(U) subset E, where U subset M. Here, there is an equivalence relation in the fiber E_x of pi over x, since the orbit of a point in fiber E_x by phi collapses onto a single point x in U via the projection map pi S^3 to S^3texttextasciitilde. After the collapse of every fiber in manifold E, the quotient space S^3S^1 is seen to be the projective complex line mathbbCP^1 cong S^2. The projective complex line is the ratio of two complex numbers. To see how the space of S^3 is connected compared to S^1, note that every closed loop in S^3 is shrinkable to a single point in a continuous way, tracing a local section. However, a closed loop in S^1 is not shrinkable to a single point. This fact makes S^3 a simply-connected space and S^1 a not simply-connected space.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image21) (Image: board12) (Image: image22)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We are now almost equipped with the tools to define a fiber bundle in a formal way. Let E F M be manifolds. The projection map pi E to M is a surjective and differentiable map (Every element in M has some element in E). Then, (E pi M F) is called a fiber bundle, (or a locally trivial fibration, or a locally trivial bundle) if for every x in M there exists an open neighborhood U subset M around the point x such that the map pi restricted to E_U can be trivialized as a cross product. Remember that E_U is the part of E of pi over U. In other words, (E pi M F) is called a fiber bundle if there exists a diffeomorphism phi_U E_U to U times F such that pr_1 o phi_U = pi, meaning the projection onto the first factor of the trivialization map phi_U is the same as the map pi. Also, a fiber bundle is denoted by F to E xrightarrowpi M. In this notation, E denotes the total space, M the base manifold, F the general fiber, pi the projection, and (U phi_U) a local trivialization or bundle chart.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image23) (Image: board13) (Image: image24)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Using a local trivialization (U phi_U) E_x = pi^-1(x) we find that the fiber E_x is an embedded submanifold of the total space E for every point x in M. Meaning, the tangent space of fiber E_x is a linear subsapce of the tangent space of E. The direct sum of the tangent subspace of the general fiber and the tangent subspace of the base manifold equals the tangent space of the total space: T_xE = V_xE bigoplus H_xE.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image25) (Image: board14) (Image: image26)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The composition of the local trivialization with the projection onto the second factor gives us yet another useful map between fibers E_x over x and the general fiber F. It is a differentiable and invertible map (diffeomorphism) and equals phi_U = pr_2 o phi_U _E_x E_x to F. Given that the local trivialization phi_U E_U to U times F is a diffeomorphism (invertible and smooth), the projection pr_1 U times F to U onto the first factor of phi_U is a submersion. That is to say the differntial of pr_1 is surjective. D pr_1 T(U times F) to TU takes vectors from the tangent space of U times F into vectors in the tangent space of U, such that every element of TU has some element in T(U times F). As a result, the map pi E to M is also a submersion, which means D pi TE to TM is surjective. Every tangent vector in the codomain TM has some tangent vector in the domain TE.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image27) (Image: board15) (Image: image28)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"So far, we have established that the bundle projection map, taking points from the total space into points in the base space pi E to M, is a submersion. For that reason, the tangent space of the base manifold M is a linear subset of the tangnet space of the total space manifold E. Now, we can use the regular value theorem for shining a light on the submersion of pi. Let a point x in M be a regular value of the smooth map pi E to M, and let the fiber E_x = pi^-1(x) be the preimage of the point x. Then, the map pi^-1 is an embedded submanifold of E of dimension dim E_x = dim E - dim M. Meaning, the tangent space of fiber E_x is a linear subspace of the tangent space of E. We can verify the result of the theorem for the Hopf bundle F to E xrightarrowpi M where dim E = 3 and dim M = 2. The regular value theorem implies that the Hopf fiber is one-dimensional, dim E_x = 3 - 2 = 1, as an embedded submanifold of the total space E. With that formal introduction we are going to sketch a visual 3D model next.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image29) (Image: board16) (Image: image30) (Image: image31)","category":"page"},{"location":"hopffibration.html#Import-the-Required-Packages","page":"Hopf Fibration","title":"Import the Required Packages","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Begin by importing a few software packages for doing algebraic operations, working with files and graphics processing units. Besides Porta, we need to use three packages: FileIO, GLMakie and LinearAlgebra. First, FileIO is the main package for IO and loading all different kind of files, including images and Comma-Separated Value (CSV) files. Second, interactive data visualizations and plotting in Julia are done with GLMakie. Finally, LinearAlgebra, as a module of the Julia programming language, provides array arithmetic, matrix factorizations and other linear algebra related functionality. However, through years of working with geometrical structures and shapes we have encapsulated certain mathematical computations and transformations into custom types and interfaces, which make up most of the functionalities of project Porta. In addition, we wrapped complicated computer graphics workflows inside custom types in order to increase the interoperability of our types with those of external packages such as GLMakie.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" import FileIO\n import GLMakie\n import LinearAlgebra\n using Porta","category":"page"},{"location":"hopffibration.html#Set-Hyperparameters","page":"Hopf Fibration","title":"Set Hyperparameters","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"There are essential hyperparameters that determine the complexity of graphics rendering as well as the position and orientation of a camera, through which we render a scene. Since the output of the model is an animation video, we need to set the figure size to 1920 by 1080 to have a full high definition window, in which the scene is located. Most of the shapes and objects that we put inside of the scene are two-dimensional surfaces. Therefore, the segmentation of most shapes requires two integer values for determining how much compute power and resolution we are willing to spend on the animation. Furthermore, the shape of a circle is the most common in our scenes because of the magic of complex numbers. It is known that using 30 segments results in smooth low-polygon circles. So for a two-dimensional sphere a 30 by 30 segmented two-surface should look good. Set the segments equal to 30, and less curvy shapes will look even better in consequence. But, an animation extends through time frame by frame and so we need to set the total number of frames. In this way, specifying the number of frames determines the length of the video. For example, 1440 frames make a one-minute video at 24 frames per second.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" figuresize = (1920, 1080)\n segments = 30\n frames_number = 1440","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A model means a complicated geometrical shape contained inside a graphical scene. Every model has a name to use as the file name of the output video. Here, we choose the name planethopf as we construct an alternative view of the Planet Hopf by Dror Bar-Natan (2010). Heinz Hopf in 1931 discovered a way to join circles over the skin of the globe. The discovery defines a fiber bundle where the base space is the spherical Earth and the fibers are circles. But, the circles are all mutually parallel and linked. Moreover, the Earth goes through a full rotation about the axis that connect the poles every 24 hours. So it is not surprising that the picture of a non-trivial bundle and the spinning of the base space coordinates (longitudes) makes for a ridiculous geometric shape. But, the surprising fact is that all of it is visualizable as a 3D object. Then, we use a dictionary that maps indices to names in order to keep track of boundary data on the globe and the name of each boundary as a sovereign country.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" modelname = \"planethopf\"\n indices = Dict()","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The Hopf fibration, as a fiber bundle, has an inner product space. The inner product space is symmetric, linear and positive semidefinite. The last property means that the product of a point in the bundle with itself is always non-negative, and it is zero if and only if the point is the zero vector. The abstract inner product space allows us to talk about the length of vectors, the distance between two points and the idea of orthogonality between two vectors. A pair of vectors are orthogonal when they make a right angle with each other and as a consequence their product is equal to zero. For all u v v_1 v_2 in V and alpha beta in R the following are the properties of the abstract inner product space:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Symmetric: u v = v u\nLinear: u alpha v_1 + beta v_2 = alpha u v_1 + beta u v_2\nPositive semidefinite: u u geq 0 for all u in V with u u = 0 if and only if u = 0","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Now, in order to skin the horizontal cross-sections of the bundle for visualization we need to start with a base point, which is denoted by x. At the tangent space of the base point q, the inner product space (characterized by a connection one-form) splits the tangent space of the bundle E at x into two linear subspaces: horizontal and vertical.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"T_q E = V_q E bigoplus H_q E","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In terms of the connection, the two subspcaes are orthogonal. A chart is a four-tuple of real numbers to be used as a pair of closed intervals in the horizontal subspace. Then, using the exponential map one can travel in both horizontal and vertical directions and cover the whole bundle within the lengths of the chart intervals. Within the boundary of the chart and with an additional vertical coordinate (a gauge) we can define a tubular neighborhood of the base point q. The first two elements of the four-tuple chart give the interval along the first basis vector and the last two elements give the interval along the second basis vector. As for the third basis vector of the tangent space (the vertical subspace) we use a beginning and an ending gauge.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"For the purpose of the construction of the Hopf fibration we define the bundle atlas of a general fiber bundle F to E xrightarrowpi M as an open covering U_i_i in I of the base manifold M together with bundle charts phi_i E_U_i to U_i times F. Putting the open covering with bundle charts a bundle atlas is denoted by U_i phi_i_i in I. The index i suggests that a bundle atlas should have more than one bundle chart whenever it is a non-trivial bundle (a twisted product rather than a Cartesian product). In order to cover the Hopf bundle we use the exponential matrix function supplied with linear combinations of elements from the Lie algebra so(4), which produces elements in the Lie group SO(4) that push a base point around the 3-sphere. As a side note, a Lie algebra is a vector space V that is equipped with the Lie bracket map sdot sdot V times V to V, with sdot sdot having three properties: bilinear, antisymmetric and satisfies the Jacobi identity. We choose a base point in the 3-sphere q in S^3 and then use Lie algebra elements before exponentiation in order to rotate the 3-sphere to cover every other point in the total space S^3 over the chart.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" q = Quaternion(ℝ⁴(0.0, 0.0, 1.0, 0.0))\n chart = (-π / 4, π / 4, -π / 4, π / 4)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, we define five scalars in the Lie algebra of so(2), identified with imathbbR, in order to provide different gauge transformations for pullbacks by the Hopf fibration (whirls and base maps). The exponential function takes the gauge values to the unit circle S^1 = U(1) cong SO(2) given by exp(im * gauge). For creating a clearer view we are going to slice up the Hopf fibers (orbits) and set different values for their respective alpha channels. The names gauge1, gauge2, gauge3, gauge4 and gauge5 are used to provide the Hopf actions when we construct and update the shapes. 0.0 means the trivial action whereas 2π means the full orbit around a Hopf fiber. Looking at the values of these names we can see that a Hopf fiber will be cut into four quarters. We can make some quarters opaque and others see-through for better visibility.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" gauge1 = 0.0\n gauge2 = π / 2\n gauge3 = float(π)\n gauge4 = 3π / 2\n gauge5 = 2π","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The fundamental physics is based on the gauge symmetry of the product SU(3) times SU(2) times U(1) and the symmetry of spacetime as a Riemannian manifold M that is equipped with a metric. Therefore, physical laws in nature must be the same under two sets of choices: the choice of gauge transformations and the choice of an inertial reference frame in spacetime. In this model, we understand the choice of the guage symmetry by studying the Hopf action and the choice of an inertial frame in Minkowski space-time by a change-of-basis transformation on the Hopf bundle. The change-of-basis transformation is denoted by matrix M and is applied to the total space of the Hopf bundle via a matrix-vector product. Here, we initialize the matrix M with the idenity.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" M = I(4)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In order to get the essence of these different choices and integrate them into a visual model we first note that Lorentz transformations of null vectors in the tangent space of spacetime is the same as transforming any other timelike (non-null) vectors. Second, The Hopf bundle of the 3-sphere has a representation in the Lie group S^3 = SU(2) and the Hopf action is represented by actions of S^1 = U(1) as a linear scalar multiplication on the right. But, null vectors have length zero in terms of the Lorentzian metric, whereas the Hopf bundle is made of vectors of unit length in terms of the Euclidean metric. Fortunately, these vectors coincide as unit quaternions and so their transformations can be unified into a single visual model. If we coordinatize a null vector in spacetime as u = 𝕍(T, X, Y, Z) then the corresponding quaternion q = Quaternion(T, X, Y, Z) takes the same coordinates. We assert that u is null and q is of unit norm, with an approximate equality check. The precision of the assertion is given by the name tolerance, which equals 1e-3.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" T, X, Y, Z = vec(normalize(ℝ⁴(1.0, 0.0, 1.0, 0.0)))\n u = 𝕍(T, X, Y, Z)\n q = Quaternion(T, X, Y, Z)\n tolerance = 1e-3\n @assert(isnull(u, atol = tolerance), \"u in not a null vector, $u.\")\n @assert(isapprox(norm(q), 1, atol = tolerance), \"q in not a unit quaternion, $(norm(q)).\")","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The camera is a viewport trough which we see the scene. It is a three-dimensional camera and much like a drone it has six features to help position and orient itself in the scene. Accordingly, a three-vector in the Euclidean 3-space E^3 determins its position in the scene, another 3-vector specifies the point at which it looks, and a third vector controls the up direction of the camera. The third 3-vector is needed because the camera can rotate through 360 degrees about the axis that connects its own position to the position of the subject. Using these three 3-vectors we control how far away we are from the subject, and how upright the subject is. ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" eyeposition = normalize(ℝ³(1.0, 1.0, 1.0)) * π * 0.8\n lookat = ℝ³(0.0, 0.0, 0.0)\n up = normalize(ℝ³(1.0, 0.0, 0.0))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Each of the eyeposition, lookat and up vectors are in the three-real-dimensional vector space ℝ³. The structure of the abstract vector space of ℝ³ includes: associativity of addition, commutativity of addition, the zero vector, the inverse element, distributivity Ι, distributivity ΙΙ, associativity of scalar multiplication, and the unit scalar 1. Also, the product space associated with ℝ³ is symmetric, linear and positive semidefinite (see real3_tests.jl). The same goes for the structure of 4-vectors in ℝ⁴ as we are going to encounter in this model. An abstract vector space (V mathbbK + ) consists of four things:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A set of vector-like objects V = u v \nA field mathbbK of scalar numbers, complex numbers, quaternions, or octonions (any one of the division algebras)\nAn addition operation + for elements of V that dictates how to add vectors: u + v\nA scalar multiplication operator for scaling a vector by an element of the field","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"An abstract vector space satisfies eight axioms. For all vectors u v w in V and for all scalars alpha beta in mathbbK the following properties are true:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Associativity of addition: u + (v + w) = (u + v) + w\nCommutativity of addition: u + v = v + u\nThere exists a zero vector 0 in V such that u + 0 = 0 + u = u\nFor every u there exists an inverse element -u such that u + (-u) = u - u = 0\nDistributivity I: alpha (u + v) = alpha u + alpha v\nDistributivity II: (alpha + beta) u = alpha u + beta u\nAssociativity of scalar multiplication: alpha (beta u) = (alpha beta) u\nThere exists a unit scalar 1 such that 1u = u","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Interestingly, if the field mathbbK is an Octonian number then the axiom of the commutativity of addition becomes false. The plan is to first load a geographic data set, then construct a few shapes, and animate a four-stage transformation of the shapes. Model versioning can be applied here using different stages. The transformations are subgroups of the Lorentz transformation in the Minkowski vector space 𝕍, which is a tetrad and origin point away from the Minkowski space-time 𝕄. Both 𝕍 and 𝕄 inherit the properties of the abstract vector space. See minkowskivectorspace_tests.jl and minkowskispacetime_tests.jl for use cases.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" totalstages = 4","category":"page"},{"location":"hopffibration.html#Load-the-Natural-Earth-Data","page":"Hopf Fibration","title":"Load the Natural Earth Data","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, we need to load two image files: an image to be used as a color reference, and another one to be used as surface texture for sections of the Hopf bundle. This is the first example of using FileIO to load image files from hard drive memory. Both images are made with a software called QGIS, which is a geographic information system software that is free and open-source. But, the data comes from Natural Earth Data. Natural Earth is a public domain map dataset available at 1:10m, 1:50m, and 1:110 million scales. Featuring tightly integrated vector and raster data, with Natural Earth you can make a variety of visually pleasing, well-crafted maps with cartography or GIS software. We downloaded the Admin 0 - Countries data file from the 1:10m Cultural Vectors link of the Downloads page. It is a large-scale map that contains geometry nodes and attributes.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" reference = FileIO.load(\"data/basemap_color.png\")\n mask = FileIO.load(\"data/basemap_mask.png\")","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"As for the image files, we paint the boundaries using the gemometry nodes, and add a grid to be able to visualize distortions of the Euclidean metric of the underlying surface. Therefore, the reference is the clean image from which we pick colors, whereas the mask has a grid and transparency for visualization purposes.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" attributespath = \"data/naturalearth/geometry-attributes.csv\"\n nodespath = \"data/naturalearth/geometry-nodes.csv\"\n countries = loadcountries(attributespath, nodespath)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The geometry nodes of the data set consist of latitudes and longitudes of boundaries. But, geometry attributes feature various geographical, cultural, economical and geopolitical values. Of these features we only need the names and geographic coordinates. To not limit the use cases of this model, the generic function loadcountries loads all of the data features by supplying it with the file paths of attributes and nodes. Data versioning can be applied here using different file versions. The attributes and nodes files are comma-separated values.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"At a high level of description, the process of loading boundary data is as follows: First, we use FileIO to open the attributes file. Second, we put the data in a DataFrames object to have in-memory tabular data. Third, sort the data according to shape identification. Fourth, open the nodes file in a DataFrame. Fifth, group the attributes by the name of each sovereign country. Sixth, determine the number of attribute groups by calling the generic function length. Seventh, define a constant ϵ = 5e-3 to limit the distance between nodes so that the computational complexity becomes more reasonable. Eighth, define a dictionary that has the keys: shapeid, name, gdpmd, gdpyear, economy, partid, and nodes. Finally, for each group of the attributes we extract the data corresponding to the dictionary keys and push them into array values.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Part of the difficulty with the data loading process is that each sovereign country may have more than one connected component (closed boundary). That is why we store part identifications as one of the dictionary keys. In this process, the part with the greatest number of nodes is chosen as the main part and is pushed into the corresponding array value. All of the array values are ordered and have the same length so that indexing over the values of more than one key becomes easier. Once the part ID of each country name is determined, we make a subset of the data frame related to the part ID and then extract the geographic coordinates in terms of latitudes and longitudes. In fact, we make a histogram of each unique part ID and count the number of coordinates. The part ID with the greatest number of coordinates is selected for creating the subset of the data frame. Next, the coordinates are transformed into the Cartesian coordinate system from the Geographic one.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Finally, we decimate a curve containing a sequence of coordinates by removing points from the curve that are farther from each other than the given threshold ϵ. It is a step to make sure that the boundary data has superb quality while managing the size of data for computation complexity. The generic function decimate implements the Ramer–Douglas–Peucker algorithm. It is an iterative end-point fit algorithm suggested by Dror Bar-Natan (2010) for this model. Since a boundary is modelled as a curve of line segments, we set a segmentation limit. But, the decimation process finds a curve that is similar in shape, yet has fewer number of points with the given threshold ϵ. In short, decimate recursively simplifies the segmented curve of a closed boundary if the maximum distance between a pair of consecutive points is greater than ϵ. The distance between two abstract vectors is given by d(u v) equiv u - v = sqrt(u - v) (u - v).","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" boundary_names = [\"United States of America\", \"Antarctica\", \"Australia\", \"Iran\", \"Canada\", \"Turkey\", \"New Zealand\", \"Mexico\", \"Pakistan\", \"Russia\"]\n boundary_nodes = Vector{Vector{ℝ³}}()\n for i in eachindex(countries[\"name\"])\n for name in boundary_names\n if countries[\"name\"][i] == name\n push!(boundary_nodes, countries[\"nodes\"][i])\n println(name)\n indices[name] = length(boundary_nodes)\n end\n end\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"As the boundary data is massive in number (248 countries) we need to select a subset for visualization. 10 countries selected from a linear space of alphabetically sroted names should be representative of the whole Earth. Then again, using only three distinct points in the 2-sphere one can infer the transformations from the sphere into itself. Also, Antarctica should be added due to its special coordinates at the south pole, to give the user a better sense of how bundle sections are expanded and distorted. As soon as we have the names of the selection, we can proceed with populating the dictionary of indices that relates the name of each country with the corresponding index in boundary data. Using the dictionary we can read the attributes of countries by giving just the name as argument.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" points = Vector{Quaternion}[]\n for i in eachindex(boundary_nodes)\n _points = Quaternion[]\n for node in boundary_nodes[i]\n r, θ, ϕ = convert_to_geographic(node)\n push!(_points, q * Quaternion(exp(ϕ / 4 * K(1) + θ / 2 * K(2))))\n end\n push!(points, _points)\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We instantiate a vector of a vector of type Quaternion to store boundary data. The outermost vector contains elements of different countries. But, the innermost vector contains the pullback of the geographic nodes by the Hopf map in the 3-sphere. After conversion to the Geographic coordinate system from the Cartesian coordinates, the points are pulled back by pi using the statement q * Quaternion(exp(ϕ / 4 * K(1) + θ / 2 * K(2))). It is a right multiplication of the base point q by the exponential function, supplied with the geographic coordinates θ and ϕ. Now that we have the points we can make a 3D scene.","category":"page"},{"location":"hopffibration.html#Make-a-Computer-Graphical-Scene","page":"Hopf Fibration","title":"Make a Computer Graphical Scene","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Scenes are fundamental building blocks of GLMakie figures. In this model, the layout of the Figure (graphical window) is a single Scene, because we have been able to directly plot all of the information about the bundle geometry and topology inside the same scene. The figure is supplied with the hyperparameter figuresize that we defined earlier. Then, we set a black theme to have black background around the window at the margins. Next, we instantiate a gray point light and a lighter gray ambient light. The lights together with the figure are then passed to LScene to construct our scene. We pass the symbol :white as the argument to the background keyword as it makes for the most visible scene.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" makefigure() = GLMakie.Figure(size = figuresize)\n fig = GLMakie.with_theme(makefigure, GLMakie.theme_black())\n pl = GLMakie.PointLight(GLMakie.Point3f(0), GLMakie.RGBf(0.0862, 0.0862, 0.0862))\n al = GLMakie.AmbientLight(GLMakie.RGBf(0.9, 0.9, 0.9))\n lscene = GLMakie.LScene(fig[1, 1], show_axis=false, scenekw = (lights = [pl, al], clear=true, backgroundcolor = :white))","category":"page"},{"location":"hopffibration.html#Construct-Base-Maps","page":"Hopf Fibration","title":"Construct Base Maps","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The base map is the pullback of the skin of the globe U subset S^2 by the Hopf map pi S^3 to S^2, representing a local horizontal cross-section of the bundle. The pushforward of horizontal vectors by the Hopf map leaves them unchanged. However, vectors in the vertical subsapce of the tangent space of the Hopf bundle are in the kernel of the Hopf map (they are sent to zero).","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We use a 64-bit floating point number to parameterize an element of the Lie algebra so(2), before exponentiating it into an element of the Lie group SO(2) to be used for the orbit map phi S^1 to S^3, because a local horizontal cross-section uses the same scalar number for the entirety of subset U subset S^2. The subset U is bounded with a two-dimensional chart. A chart can be thought of as a rectangle whose sides are at most π in length. But, the length of a great circle of the three-dimensional sphere is 2π and the maximum length of chart sides is limited, unless we want to cover S^3 twice. To keep things simple, we use one bundle chart and cover a subset U of side length π. The Hopf bundle does not admit a global section. After exponentiating the base point q in horizontal directions for a magnitude beyond π, the orientation of the surface reverses and a sharp twist of the surface happens.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The Hopf bundle is embedded in ℝ⁴, the real-four-dimensional space. The coordinates are defined as unit quaternions where the basis vectors are represented by the symmetry group of the rotations of an orthogonal tetrad, namely SO(4). vectors u and v are orthogonal if and only if their inner product equals zero u v = 0. When we talk about Hopf actions and bundle charts, we talk about values that are used to linearly combine elements of the Lie algebra of so(4), vectors in the tangent space of the bundle at point x. Then, we use the matrix exponential map for computing Lie group values in SO(4). Given a fixed gauge, a point in the Lie group stemming from base point x is reconstructed from a Lie algebra element by executing the statement x * Quaternion(exp(θ * K(1) + -ϕ * K(2)) * exp(gauge * K(3))), where scalars θ and ϕ denote the latitude and longitude components in the bundle chart, respectively. K(1) and K(2) denote 4x4 matrices with real elements as basis vectors of the Lie algebra so(4). The tangent space of the bundle at point x spans horizontally with the exponential map of a linear combination of basis vectors K(1) and K(2), whereas it spans vertically in the K(3) direction. This way we get a strictly horizontal section of the bundle in terms of elements of the Lie group SO(4), given a gauge. The elements of SO(4) go on to push the base point x around and end up as observables to be rendered graphically.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" lspaceθ = range(chart[1], stop = chart[2], length = segments)\n lspaceϕ = range(chart[3], stop = chart[4], length = segments)\n [project(normalize(M * (x * Quaternion(exp(θ * K(1) + -ϕ * K(2)) * exp(gauge * K(3)))))) for ϕ in lspaceϕ, θ in lspaceθ]","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Using the eigendecomposition method LinearAlgebra.eigen, we can compute the matrix M to change the basis of the bundle while keeping the coordinates invariant. So the change-of-basis is the final step of the construction of the observables after using the geographic coordinates and the gauge. Observables.jl allows us to define the points that are to be rendered in the scene, in a way that they can listen to changes dynamically. Later, when we apply transformations to the bundle, including the change-of-basis, the idea is to only change the top-level observables and avoid reconstructing the scene entirely. The change of basis is a bilinear transformation of the tetrad (of Minkowski space-time 𝕄) in ℝ⁴ as a matrix-vector product (M * x for example). Here we denote the transformation as matrix M, which takes a Quaternion number as input and spits out a new number of the same type. The input and output bases must be orthonormal as the numbers must remain unit quaternions after the transformation. Constructing a base map requires a few arguments: the scene object, the base point q, the gauge, the change-of-basis transformation M, the chart, the number of segments of the lattice of observables, the tuxture of the surface and the optional transparency setting. Construct four base maps in order to visualize a more complete picture of the Hopf fibration using four different sections. But, the sections are going to be distinguished from one another and updated with gauge transformations later when we animate them.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" basemap1 = Basemap(lscene, q, gauge1, M, chart, segments, mask, transparency = true)\n basemap2 = Basemap(lscene, q, gauge2, M, chart, segments, mask, transparency = true)\n basemap3 = Basemap(lscene, q, gauge3, M, chart, segments, mask, transparency = true)\n basemap4 = Basemap(lscene, q, gauge4, M, chart, segments, mask, transparency = true)","category":"page"},{"location":"hopffibration.html#Construct-Whirls","page":"Hopf Fibration","title":"Construct Whirls","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A Whirl is the shape of a closed boundary in the map of the Earth that is pulled back by the Hopf map pi S^3 to S^2. As a reminder, boundaries on the map of the Earth are specified by two real values: latitude θ and longitude ϕ. The boundary of each country in boundary_names is lifted up from the base manifold using the following statement: q * Quaternion(exp(ϕ / 4 * K(1) + θ / 2 * K(2))). The pullback operation is realized by pushing the base point q in a horizontal direction given by coordinates on the surface of the Earth. Then, a gauge transformation is applied by executing the statement x * Quaternion(exp(K(3) * gauge)), with the given scalar gauge in the direction K(3) of the tangent space at point x of the bundle. By varying gauge in a linear space of floating point values, a Whirl (a pullback by the Hopf map) takes a three-dimensional volume. In the special case where gauge is a range of values, starting at zero and stopping at 2π, the Whirl makes a Hopf band. The degree of the twist in the band is directly proportional to the value of gauge. Multiplying x on the right by the exponentiation of K(3) * gauge pushes x in the vertical subspace of the bundle and makes an orbit. Therefore, the orbit map phi S^1 to S^3 is given by x[i] * Quaternion(exp(K(3) * gauge).","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" lspacegauge = range(gauge1, stop = gauge2, length = segments)\n [project(normalize(M * (x[i] * Quaternion(exp(K(3) * gauge))))) for i in 1:length(x), gauge in lspacegauge]","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"There are four sets of whirls: some whirls are more solid and some whirls are more transparent. This separation is done to highlight the antipodal points of the three-dimensional sphere S^3, given by x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1, where x_1 x_2 x_3 x_4^T in R^4. It also helps to visualize the direction of the null plane under transformations of the bundle. Since every pair of points that are infinitestimally close to each other in a horizontal cross-section, defines a differential operator. And Hopf actions, transformations from the bundle into itself change the direction of the operator as it twists. The operator is also called a spin-vector in Minkowski vector space 𝕍. Therefore it can be visualized directly how the operator changes sign by comparing a pullback into S^3 at antipodal points of an orbit.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" whirls1 = []\n whirls2 = []\n whirls3 = []\n whirls4 = []\n for i in eachindex(boundary_nodes)\n color1 = getcolor(boundary_nodes[i], reference, 0.1)\n color2 = getcolor(boundary_nodes[i], reference, 0.2)\n color3 = getcolor(boundary_nodes[i], reference, 0.3)\n color4 = getcolor(boundary_nodes[i], reference, 0.4)\n whirl1 = Whirl(lscene, points[i], gauge1, gauge2, M, segments, color1, transparency = true)\n whirl2 = Whirl(lscene, points[i], gauge2, gauge3, M, segments, color2, transparency = true)\n whirl3 = Whirl(lscene, points[i], gauge3, gauge4, M, segments, color3, transparency = true)\n whirl4 = Whirl(lscene, points[i], gauge4, gauge5, M, segments, color4, transparency = true)\n push!(whirls1, whirl1)\n push!(whirls2, whirl2)\n push!(whirls3, whirl3)\n push!(whirls4, whirl4)\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The color of a Whirl should match the color of the inside of its own boundary at every horizontal section, also known as a base map. The generic function getcolor finds the correct color to set for the Whirl. It takes as input a closed boundary (a vector of Cartesian points), a color reference image and an alpha channel value to produce an RGBA 4-color. getcolor finds a color according to the following steps: First, it determines the number of points in the given boundary. Second, gets the size of the reference color image as height and width in pixels. Third, converts all of the boundary points to Geographic coordinates. Fourth, finds the minimum and maximum values of the latitudes and longitudes of the boundary. Fifth, creates a two-dimensional linear space (a flat grid or lattice) that ranges within the upper and lower bounds of the latitudes and longitudes. Sixth, finds the Cartesian two-dimensional coordinates of the points in the image space by normalizing the geographic coordinates and multiplying them by the image size. Seventh, picks the color of each grid point with the Cartesian two-dimensional coordinates in the image space as the index. Eighth, Makes a histogram of the colors by counting the number of each color. Finally, sorts the histogram and picks the color with the greatest number of occurance. (See earth.jl from the src directory for implementation.)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"However, step seven makes sure that the coordinates in the linear 2-space are inside the closed boundary, otherwise it skips the index and continues with the next index in the grid. In this way we don't pick colors from the boundaries of neighboring countries over the globe. The generic function isinside is used by getcolor to determine whether the given point is inside the given boundary or not. But first, the boundary needs to become a polygon in the Euclidean 2-space of coordinates in terms of latitude and longitude. This is the same as geographic coordinates with the radius of Earth set equal to 1 identically, hence the spherical Earth model of the ancient Greeks. After we make a polygon out of the boundary, the generic function rayintersectseg determines whther a ray cast from a point of the linear grid intersects an edge with the given point p and edge. Here, p is a two-dimensional point and edge is a tuple of such points, representing a line segment. Eventhough this algorithm should work in theory, some boundaries are too small to yield a definite color via getcolor and the color inference algorithm returns a false negative in those cases. So the default color may be white for a limited number of cases out of 248 countries. Once we have the color of the whirls, we can proceed to construct the whirls by supplying the generic function Whirl with the following arguments: the scene object, the boundary points lifetd via an arbitrary section, the first fiber action value (gauge), the second action value, the change-of-basis function M, the number of surface segments, the color and the optional transparency setting.","category":"page"},{"location":"hopffibration.html#Compute-a-Four-Screw","page":"Hopf Fibration","title":"Compute a Four-Screw","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We are going to execute a motion around a closed loop in the Lie group SL(2 mathbbC), and then multiply every point in the Hopf bundle by an element of the loop. A four-screw is a subset of the Lie group SL(n mathbbK) = A in Mat(n times n mathbbK) det(A) = 1, square matrices of Complex numbers whose volume form (determinant) equals 1. Here, the number n = 2 and the field mathbbK = mathbbC. A four-screw is a kind of restricted Lorentz transformation where a z-boost and a proper rotation of the celestial sphere are applied. The transformation lives in a four-complex dimensional space and it has six degrees of freedom (the same number of dimensions as SO(4)). By parameterizing a four-screw one can control how much boost and rotation a transformation shuld have. Here, w as a positive scalar controls the amount of boost, whereas angle ψ controls the rotation component of the transform. But, the parameterization accepts rapidity as input for the boost. So we take the natural logarithm of w (log(w) = phi) in order to supply the transformer with the required rapidity argument. First, we set w equal to one in order to preserve the scale of the Argand plane and animate the angle ψ through zero to 2π for rotation. The name progress denotes a scalar from zero to one for instantiating a different transformation at each frame of the animation.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" if status == 1 # roation\n w = 1.0\n ϕ = log(w) # rapidity\n ψ = progress * 2π\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In the second case, we fix the rotation angle ψ by setting it to zero, and this time animate the rapidity by changing the value of ϕ at each time step.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" if status == 2 # boost\n w = max(1e-4, abs(cos(progress * 2π)))\n ϕ = log(w) # rapidity\n ψ = 0.0\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Third, in order to get a complete picture of a four-screw we animate both rapidity ϕ and rotation ψ, at the same time.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" if status == 3 # four-screw\n w = max(1e-4, abs(cos(progress * 2π)))\n ϕ = log(w) # rapidity\n ψ = progress * 2π\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A four-real-dimensional vector in the Minkowski vector space 𝕍 is null if and only if its Lorentz norm is equal to zero. The length or norm of an abstract vector u in V is equivalent to the square root of the inner product of the vector with itself: u u equiv sqrtu u in R. The inner product of vectors u and v in an abstract vector space is given by u^T * g_munu * v, where g_munu denotes the metric 2-tensor. However, as an instantiation in Minkowski vector space 𝕍 with signature (+, -, -, -), the matrix g_munu is a diagonal of the form: g_munu = beginbmatrix 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 endbmatrix.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Furthermore, a vector in 𝕍 is in the tangent space at some point in Einstein's spacetime, where the metric g_munu will not be diagonal in general. Since a Lorentz transformation of null vectors has the same effect on vectors that are not null, it makes the visualization easier to study transformations on null vectors only. On the other hand, in the Euclidean 4-space E^4 the metric g_munu is replaced by identity matrix of dimension four. The null vectors that we use here in the Minkowski vector space have length zero in terms of the Lorentz norm, but have Euclidean norm equal to one, and so they can be regarded as elements of unit Quaternion. Therefore, what we are animating here is the transformation of unit quaternions that represent null vectors. ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The change-of-basis transformations that we have used to instantiate Whirl and Basemap types above, can accomodate the effects of a Lorentz transformation. Then, by setting ψ and ϕ we can define a generic function transform to take Quaternion numbers as input and to give us the transformed number as output.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" transform(x::Quaternion) = begin\n T, X, Y, Z = vec(x)\n X̃ = X * cos(ψ) - Y * sin(ψ)\n Ỹ = X * sin(ψ) + Y * cos(ψ)\n Z̃ = Z * cosh(ϕ) + T * sinh(ϕ)\n T̃ = Z * sinh(ϕ) + T * cosh(ϕ)\n Quaternion(T̃, X̃, Ỹ, Z̃)\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Every transformation in an abstract vector space such as the Minkowski vector space 𝕍 has a matrix representation. For constructing the matrix of the transform we just need to compute it four times with basis vectors. The transformation of the basis vectors of unit quaternions by transform are denoted by r₁, r₂, r₃ and r₄. The matrix _M is a four by four real matrix whose rows are r₁ through r₄. _M is the matrix representation of the transformation that is induced by transform.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" r₁ = transform(Quaternion(1.0, 0.0, 0.0, 0.0))\n r₂ = transform(Quaternion(0.0, 1.0, 0.0, 0.0))\n r₃ = transform(Quaternion(0.0, 0.0, 1.0, 0.0))\n r₄ = transform(Quaternion(0.0, 0.0, 0.0, 1.0))\n _M = reshape([vec(r₁); vec(r₂); vec(r₃); vec(r₄)], (4, 4))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"But, _M doesn't necessarily take unit quaternions to unit quaternions. By decomposing _M into eigenvalues and eigenvectors we can manipulate the transformation so that it takes unit quaternions to unit quaternions without modifying its effect on the geometrical structure of Argand plane. Despite the fact that _M is a matrix of real numbers, it has complex eigenvalues, as it involves a rotation. By constructing a four-complex-dimensional vector off of the eigenvalues we can normalize _M by normalizing the vector of eigenvalues, before reconstructing a unimodular, unitary transformation (a normal matrix). The reconstructed matrix is called M.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" decomposition = LinearAlgebra.eigen(_M)\n λ = LinearAlgebra.normalize(decomposition.values) # normalize eigenvalues for a unimodular transformation\n Λ = [λ[1] 0.0 0.0 0.0; 0.0 λ[2] 0.0 0.0; 0.0 0.0 λ[3] 0.0; 0.0 0.0 0.0 λ[4]]\n M = real.(decomposition.vectors * Λ * LinearAlgebra.inv(decomposition.vectors))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We can assert that the transformation that is induced by M takes null vectors to null vectors in Minkowski vector space 𝕍. If that is the case, then the reconstructed transformation M is a faithful representation and it only scales the extent of null vectors rather than null directions, compared to _M. A representation f is called a faithful representation when for different numbers g and q, f(g) and f(q) are equal if and only if g = q.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A spin-vector is based on the space of future or past null directions in Minkowski space-time. The field ζ of a SpinVector represents points in Argand plane. Therefore, if v is obtained with the transformation of u by M, then the respective spin-vectors s and s′ should tell us how M changes Argand plane. To be precise, three different points in Argand plane, namely u₁, u₂, u₃, are needed to characterize the transformation. We assert that the transformation by M induced on Argand plane is correct, because it extends the Argand plane ζ = w * exp(im * ψ) * s.ζ by magnitude w and rotates it through angle ψ. So, we established the fact that normalizing the vector of eigenvalues of the transformation _M and reconstructing it to get M leaves the effect on Argand plane invariant.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" u₁ = 𝕍(1.0, 1.0, 0.0, 0.0)\n u₂ = 𝕍(1.0, 0.0, 1.0, 0.0)\n u₃ = 𝕍(1.0, 0.0, 0.0, 1.0)\n for u in [u₁, u₂, u₃]\n v = 𝕍(vec(M * Quaternion(u.a)))\n @assert(isnull(v, atol = tolerance), \"v ∈ 𝕍 in not null, $v.\")\n s = SpinVector(u)\n s′ = SpinVector(v)\n if s.ζ == Inf # A Float64 number (the point at infinity)\n ζ = s.ζ\n else # A Complex number\n ζ = w * exp(im * ψ) * s.ζ\n end\n ζ′ = s′.ζ\n if ζ′ == Inf\n ζ = real(ζ)\n end\n @assert(isapprox(ζ, ζ′, atol = tolerance), \"The transformation induced on Argand plane is not correct, $ζ != $ζ′.\")\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A distinction between coordinates in Argand plane becomes relevant when we want to assert the properties of M on a test variable ζ, without applying M on a control variable ζ′. In the special case where the null direction ζ is the point at infinity, the north pole, we expect for the transformation induced by M to be inconsequential. Because ζ is a union of complex numbers and the singleton of infinity (of type Union{Complex, ComplexF64, Float64}). For an inhomogeneous coordinate system we treat the point at infinity in a different way. For example, for all values of w, if ζ equals infinity then the rotation component of a four-screw should not have any effect on the north pole. But, multiplying positive infinity by a complex number of negative magnitude makes ζ equal to negative infinity, which is not in Argand plane. In that case, we first check the edge case to leave ζ unchanged whenever its value is infinity, ζ = s.ζ. No amount of z-boost and rotation about the z-axis should transform the north pole. Else, ζ transforms as expected: ζ = w * exp(im * ψ) * s.ζ.","category":"page"},{"location":"hopffibration.html#Compute-a-Null-Rotation","page":"Hopf Fibration","title":"Compute a Null Rotation","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"To understand a null rotation, imagine that you are an astronaut in empty space, far away from any celestial object. Looking at the space around you from every direction, you can see your surrounding environment through a spherical viewport. This view is called the celestial sphere of past null directions, as the light from the stars in the past reach your eyes. A null rotation translates Argand plane such that just one null direction is invariant, the point at infinity (the north pole of the celestial sphere). We control the animation of a null rotation by defining a real number a.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" a = sin(progress * 2π)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Whenever T is positive, we talk about the sphere of future-pointing null directions. At this stage of the animation, the transformation transform defines a null rotation such that the invariant null vector is the direction t + z, the north pole of the sphere of future-pointing null directions, where ζ equals infinity. ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" transform(x::Quaternion) = begin\n T, X, Y, Z = vec(x)\n X̃ = X \n Ỹ = Y + a * (T - Z)\n Z̃ = Z + a * Y + 0.5 * a^2 * (T - Z)\n T̃ = T + a * Y + 0.5 * a^2 * (T - Z)\n Quaternion(T̃, X̃, Ỹ, Z̃)\n end\n\n r₁ = transform(Quaternion(1.0, 0.0, 0.0, 0.0))\n r₂ = transform(Quaternion(0.0, 1.0, 0.0, 0.0))\n r₃ = transform(Quaternion(0.0, 0.0, 1.0, 0.0))\n r₄ = transform(Quaternion(0.0, 0.0, 0.0, 1.0))\n _M = reshape([vec(r₁); vec(r₂); vec(r₃); vec(r₄)], (4, 4))\n decomposition = LinearAlgebra.eigen(_M)\n λ = decomposition.values\n Λ = [λ[1] 0.0 0.0 0.0; 0.0 λ[2] 0.0 0.0; 0.0 0.0 λ[3] 0.0; 0.0 0.0 0.0 λ[4]]\n M = real.(decomposition.vectors * Λ * LinearAlgebra.inv(decomposition.vectors))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, we instantiate another spin-vector using M * u = v in order to examine the effect of the transformation M on Argand plane. Specifically, the point ζ from the Argand plane of u transforms into α * s.ζ + β, where α determines the extension of Argand plane and β the translation. The scalar a controls the translation of the plane because β is defined as β = Complex(im * a). We assert that the transformation induced on Argand plane is correct by comparing the approximate equality of the Argand plane of v and the Argand plane of u. Similar to previous animation stages, the induced transformation on Argand plane by M is completely characterized using three different points: u₁, u₂, u₃. After transforming u by M we assert that the result v is still a null vector.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" u₁ = 𝕍(1.0, 1.0, 0.0, 0.0)\n u₂ = 𝕍(1.0, 0.0, 1.0, 0.0)\n u₃ = 𝕍(1.0, 0.0, 0.0, 1.0)\n for u in [u₁, u₂, u₃]\n v = 𝕍(vec(M * Quaternion(u.a)))\n @assert(isnull(v, atol = tolerance), \"v ∈ 𝕍 in not a null vector, $v.\")\n s = SpinVector(u) # TODO: visualize the spin-vectors as frames on S⁺\n s′ = SpinVector(v)\n β = Complex(im * a)\n α = 1.0\n ζ = α * s.ζ + β\n ζ′ = s′.ζ\n if ζ′ == Inf\n ζ = real(ζ)\n end\n @assert(isapprox(ζ, ζ′, atol = tolerance), \"The transformation induced on Argand plane is not correct, $ζ != $ζ′.\")\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Finally, we also assert that the null direction z + t is invariant under the transformation M because it is a null rotation with a fixed null direction at the north pole. The animation of a null rotation is correct if all of the assertions evaluate true.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" v₁ = 𝕍(normalize(ℝ⁴(1.0, 0.0, 0.0, 1.0)))\n v₂ = 𝕍(vec(M * Quaternion(vec(v₁))))\n @assert(isnull(v₁, atol = tolerance), \"vector t + z in not null, $v₁.\")\n @assert(isapprox(v₁, v₂, atol = tolerance), \"The null vector t + z is not invariant under the null rotation, $v₁ != $v₂.\")","category":"page"},{"location":"hopffibration.html#Update-the-Camera","page":"Hopf Fibration","title":"Update the Camera","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The 3D camera of the scene requires the eye position, look at, and up vectors for positioning and orientation. The function update_cam! takes the scene object along with the three required vectors as arguments and updates the camera. But, our camera position and orientation vectors are of type ℝ³, and not Vec3f. To match the argument type we need to use the generic function vec and the splat operator in order to instantiate objects of type Vec3f, because update_cam! is going to match the given type with its own signature.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" GLMakie.update_cam!(lscene.scene, GLMakie.Vec3f(vec(eyeposition)...), GLMakie.Vec3f(vec(lookat)...), GLMakie.Vec3f(vec(up)...))","category":"page"},{"location":"hopffibration.html#Record-an-Animation","page":"Hopf Fibration","title":"Record an Animation","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Updating the base maps requires a base point in the section denoted by q and the transformation M. Then, we use M to update base maps 1, 2, 3 and 4. For we want to have different choices of an inertial reference frame in the tangent space of some point in spacetime. The generic function update! updates base maps by changing the structurally embedded observables, and then the graphical shapes take different forms accordingly.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Although we are talking about points in the bundle, embedded in ℝ⁴ and of type Quaternion, the generic function project converts them to points in ℝ³. The one method of project takes the given point q ∈ S³ ⊂ ℂ² and turns it into a point in the Euclidean space E³ ⊂ ℝ³ using stereographic projection. We identify mathbbR^4 to mathbbC^2 given by (x_1 x_2 x_3 x_4) mapsto (x_1 + i x_2 x_3 + i x_4). Then, the stereographic projection is given by: project S^3 setminus (1 0) to mathbbR^3 given by (x_1 x_2 x_3 x_4) mapsto fracx_2 x_3 x_4^T1 - x_1.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Whenever we call the update! function with an object like basemap1, giving transformation M, two things happn under the hood for deforming the graphics (update!(basemap1, q, gauge1, M)). First, a matrix of type ℝ³ is made, Matrix{ℝ³}. That is the job of one of the methods of the generic function make. The correct dispatch is selected automatically for the job, based on the argument signature (whether the first argument is of type Whirl or Basemap for example). The selected method makes a 2-surface (lattice) of the horizontal section at base point q after transforming by M, with the given segments number, gauge and chart. A chart and a gauge play the role of a choice of local trivialization of the Hopf bundle, as an atlas, for the purpose of constructing a pullback of the Earth's surface.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Second, the matrix of ℝ³ along with the given basemap's observables are passed to the function updatesurface! for updating the observables. For each coordinate component x, y and z in the Euclidean 3-space E^3, there is a corresponding matrix of real numbers, of the same size: (segments by segments). In the type structure of a Basemap or a Whirl there is a tuple whose elements are of type Observable. Each element of the three-tuple in turn contains a matrix of components x, y or z. Reshaping a matrix of 3-vectors into three matrices of scalars is done because when we implicitly instantiated a GLMakie surface in the beginning, we supplied it with three observables representing x, y and z coordinates separately. The generic function buildsurface from the source file surface.jl builds a surface with the given scene, value, color and transparency. Here, the value argument is of type Matrix{ℝ³}. The interface between the construction of our base maps (or whirls) and the graphics engine is essentially a reshaping and type conversion. See surface_tests.jl for use cases.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Every time we update the observables of a Whirl under transformation by M, we need to access the coordinates of the boundary data (update!(whirls1[i], points[i], gauge1, gauge2, M)). But the coordinates are not changed, and instead the change-of-basis is taken care of by the map M. The coordinate component ϕ is divided by a factor of four since in geographic coordinates longitudes range from -π to +π, whereas latitudes range from -π / 2 to +π / 2 (exp(ϕ / 4 * K(1) + θ / 2 * K(2)))). This division rescales the longitude component of coordinates and allows us to have a square bundle chart, compared to coordinate components θ. Rescaling θ and ϕ aligns the boundaries of horizontal and vertical subspaces. We finish the animation of one time-step after updating the last Whirl.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The function animate takes as input an integer called frame and updates the scene observables according to the stages that we described earlier. First, it calculates the progress of the animation frames, dividing frame by frames_number. For different properties of Lorentz transformations we have four stages, each stage having its own progress. The signature of the four-screw animator function is compute_fourscrew(progress::Float64, status::Int). For example, stage one animates a proper rotation of Argand plane by calling the function compute_fourscrew with status equal to 1. Stage 2 animates a pure z-boost. Then, stage 3 animates a four-screw. Finally, stage 4 animates a null rotation by calling the function compute_nullrotation. After calling each stage function, we update the camera by calling the function updatecamera.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" animate(frame::Int) = begin\n progress = frame / frames_number\n stage = min(totalstages - 1, Int(floor(totalstages * progress))) + 1\n stageprogress = totalstages * (progress - (stage - 1) * 1.0 / totalstages)\n println(\"Frame: $frame, Stage: $stage, Total Stages: $totalstages, Progress: $stageprogress\")\n if stage == 1\n M = compute_fourscrew(stageprogress, 1)\n elseif stage == 2\n M = compute_fourscrew(stageprogress, 2)\n elseif stage == 3\n M = compute_fourscrew(stageprogress, 3)\n elseif stage == 4\n M = compute_nullrotation(stageprogress)\n end\n update!(basemap1, q, gauge1, M)\n update!(basemap2, q, gauge2, M)\n update!(basemap3, q, gauge3, M)\n update!(basemap4, q, gauge4, M)\n for i in eachindex(whirls1)\n update!(whirls1[i], points[i], gauge1, gauge2, M)\n update!(whirls2[i], points[i], gauge2, gauge3, M)\n update!(whirls3[i], points[i], gauge3, gauge4, M)\n update!(whirls4[i], points[i], gauge4, gauge5, M)\n end\n updatecamera()\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"To create an animation you need to use the record function. In summary, we instantiated a Scene inside a Figure. Next, we created and animated observables in the scene, on a frame by frame basis. Now, we record the scene by passing the figure fig, the file path of the resulting video, and the range of frame numbers to the record function. The frame is incremented by record and the frame number is passed to the function write to animate the observables. Once the frame number reaches the total number of animation frames, recording is finished and a video file is saved on the hard drive at the file path: gallery/planethopf.mp4.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" GLMakie.record(fig, joinpath(\"gallery\", \"$modelname.mp4\"), 1:frames_number) do frame\n animate(frame)\n end","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Description = \"News Report\"","category":"page"},{"location":"newsreport.html#Lede","page":"News Report","title":"Lede","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Show a small piece of the story.","category":"page"},{"location":"newsreport.html#Context","page":"News Report","title":"Context","text":"","category":"section"},{"location":"newsreport.html#Where,-Who,-What,-How-and-Why","page":"News Report","title":"Where, Who, What, How and Why","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Tell us the facts of the story.","category":"page"},{"location":"newsreport.html#Where","page":"News Report","title":"Where","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: fourscrew1)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"What is the relation between a spin-frame and a Minkowski tetrad? The spin-frame is denoted by omicron (black) and iota (silver). Omicron and iota serve as two flag poles, where we also show their respective flags. In order to see the flags, find the arcs in the x direction that move with omicron and iota during a series of transformations. The spin-frame is in a vector space over complex numbers. The spin space has the axioms of an abstract vector space. But, we have defined a special inner product for 2-spinors, such that the product of omicron and iota equals unity, whereas the product of iota and omicron equals minus unity. In other words, the inner product eats a pair of spin-vectors in the Hopf bundle and spits out a complex number (a scalar).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: fourscrew2)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The axes t (red), x (green), y (blue), and z (orange) are parts of a Minkowski tetrad in Minkowski spacetime. Choosing the default Minkowski tetrad, the tetrad aligns with the Cartesian axes of real dimension four. But, when we apply a spin-transformation, the tetrad no longer aligns with Cartesian coordinates, and with it the spin-frame bases omicron and iota change as well. The kinds of spin transformation that we apply are four-screws and null rotations, and so they are restricted transformations. Restricted transformations do not alter the sign of time. Here, the time sign is negative one, which is the same as the wall clock time.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: fourscrew3)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"If you look closely, there are two spheres in the middle that change hue over time. One of them is the past null cone and the other is the sphere S^{-1}. You will recognize the null cone as soon as it turns into a cone momentarily. If a spin-vector is in S^{-1}, then under restricted spin-transformations it does not leave the sphere S^{-1} to S^{+1}. The past null cone is the directions of light that reach our eyes from the past. But, the sphere S^{-1} is the light that we can observe around us in the present moment (assume we’re in deep space and away from heavenly objects). Under spin-transformations the null cone and the sphere S^{-1} change too, because they are embedded in Minkowski spacetime of dimension 4.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: nullrotation)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"timesign = -1\nο = SpinVector([Complex(1.0); Complex(0.0)], timesign)\nι = SpinVector([Complex(0.0); Complex(1.0)], timesign)\n@assert(isapprox(dot(ο, ι), 1.0), \"The inner product of spin vectors $ι and $ο is not unity.\")\n@assert(isapprox(dot(ι, ο), -1.0), \"The inner product of spin vectors $ι and $ο is not unity.\")","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"generate() = 2rand() - 1 + im * (2rand() - 1)\nκ = SpinVector(generate(), generate(), timesign)\nϵ = 0.01\nζ = Complex(κ)\nζ′ = ζ - 1.0 / √2 * ϵ / κ.a[2]\nκ = SpinVector(ζ, timesign)\nκ′ = SpinVector(ζ′, timesign)\nω = SpinVector(generate(), generate(), timesign)\nζ = Complex(ω)\nζ′ = ζ - 1.0 / √2 * ϵ / ω.a[2]\nω = SpinVector(ζ, timesign)\nω′ = SpinVector(ζ′, timesign)\n@assert(isapprox(dot(κ, ι), vec(κ)[1]), \"The first component of the spin vector $κ is not equal to the inner product of $κ and $ι.\")\n@assert(isapprox(dot(κ, ο), -vec(κ)[2]), \"The second component of the spin vector $κ is not equal to minus the inner product of $κ and $ο.\")\n@assert(isapprox(dot(ω, ι), vec(ω)[1]), \"The first component of the spin vector $ω is not equal to the inner product of $ω and $ι.\")\n@assert(isapprox(dot(ω, ο), -vec(ω)[2]), \"The second component of the spin vector $ω is not equal to minus the inner product of $ω and $ο.\")","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"t = 𝕍(1.0, 0.0, 0.0, 0.0)\nx = 𝕍(0.0, 1.0, 0.0, 0.0)\ny = 𝕍(0.0, 0.0, 1.0, 0.0)\nz = 𝕍(0.0, 0.0, 0.0, 1.0)\nο = √2 * (t + z)\nι = √2 * (t - z)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct360)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The phase of the inner product of spin-vectors is shown as a prism arc. In a Minkowski tetrad with bases t, x, y and z, (with signature (+,-,-,-)) there are a pair of basis vectors for spin-vectors: omicron and iota. For example, the spin-vectors kappa and omega, each are linear combinations of omicron and iota. The product of kappa and omega is a complex number that has a magnitude and a phase. Being spin-vectors, the arrows of omicron, iota, kappa and omega represent the flagpoles, and the flag planes are attached to the flagpoles as arcs.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct720)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In order to find the inner product of kappa and omega we make use of both flagpoles and flag planes. First, note that the flagpoles span a 2-plane in the Minkowski vector space. Then, we perform the Gram-Schmidt orthogonalization method to find the orthogonal complement of the 2-plane. Next, find the intersection of the flag planes and the orthogonal complement 2-plane from the previous step. By this step, the flag plane of kappa results in vector U, whereas the flag plane of omega projects to arrow V. Then, we normalize U and V. Finally, the angle that U and V make with each other measure pi plus two times the argument of the inner product of kappa and omega.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct1080)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Furthermore, the arrow that is denoted by p bisects the angle between U and V, and measures the phase angle minus pi half (modulus two pi). Also, a spatial rotation about the axis p is done for animating the Minkowski vector space so that all of the components of the inner product are visible from a 720-degree view.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct1440)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"A spin-vector is named kappa and another spin-vector is named omega. The extra piece of information that makes spinors special is the flagpoles of spin-vectors. Using a differential operator in the plane of complex numbers, starting with zeta complex, the spin counterpart of the spin vector zeta prime equals zeta minus one over the square root of two times a constant named epsilon, over eta (the second component of the spin-vector). Except for this transformation of zeta to zeta prime, which is parameterized by epsilon, the spin-vectors kappa and kappa prime have the same features such as time sign. The same transformation produces the names omega and omega prime. With iota and omicron as the basis vectors of the spin-space G dot, we assert the following propositions:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The first component of the spin vector κ is equal to the inner product of κ and ι.\nThe second component of the spin vector κ is equal to minus the inner product of κ and ο.\nThe first component of the spin vector ω is equal to the inner product of ω and ι.\nThe second component of the spin vector ω is equal to minus the inner product of ω and ο.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"κ = 𝕍(κ)\nκ′ = 𝕍(κ′)\nω = 𝕍(ω)\nω′ = 𝕍(ω′)\nzero = 𝕍(0.0, 0.0, 0.0, 0.0)\nB = stack([vec(κ), vec(ω), vec(zero), vec(zero)])\nN = LinearAlgebra.nullspace(B)\na = 𝕍(N[begin:end, 1])\nb = 𝕍(N[begin:end, 2])\na = 𝕍(LinearAlgebra.normalize(vec(a - κ - ω)))\nb = 𝕍(LinearAlgebra.normalize(vec(b - κ - ω)))","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"When we stack the Minkowski vector space representation of kappa and omega and fill the rest with zero to get a square matrix B, the null space of B is where the piece of information about spinors exist. By performing a Gram-Schmidt procedure we find the set of orthonormal basis vectors for the inner product of kappa and omega. In the following lines, the spin-vectors an and b are bases of the null space of matrix B.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"v₁ = κ.a\nv₂ = ω.a\nv₃ = a.a\nv₄ = b.a\n\ne₁ = v₁\nê₁ = normalize(e₁)\ne₂ = v₂ - dot(ê₁, v₂) * ê₁\nê₂ = normalize(e₂)\ne₃ = v₃ - dot(ê₁, v₃) * ê₁ - dot(ê₂, v₃) * ê₂\nê₃ = normalize(e₃)\ne₄ = v₄ - dot(ê₁, v₄) * ê₁ - dot(ê₂, v₄) * ê₂ - dot(ê₃, v₄) * ê₃\nê₄ = normalize(e₄)\n\nê₁ = 𝕍(ê₁)\nê₂ = 𝕍(ê₂)\nê₃ = 𝕍(ê₃)\nê₄ = 𝕍(ê₄)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproductspositiveus)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The flag planes of kappa and omega are obtained by subtracting kappa from kappa prime and omega from omega prime, respectively. Projecting the flag plane of kappa onto the 2-plane spanned by subspace bases of ê₃ and ê₄ gives you vector U. The same subspace gives you V for the flag plane of omega. The inner product eats two spin-vectors such as kappa and omega, and spits out a complex number that has a magnitude and a phase angle. The angle that U and V make with each other determines the phase of the inner product times two plus pi. This 2-plane is the orthogonal complement of the 2-plane that contains kappa and omega (and is spanned by ê₁ and ê₂). The camera looks at the sum of the vectors kappa and omega.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"κflagplanedirection = 𝕍(LinearAlgebra.normalize(vec(κ′ - κ)))\nωflagplanedirection = 𝕍(LinearAlgebra.normalize(vec(ω′ - ω)))\nglobal u = LinearAlgebra.normalize(vec((-dot(ê₃, κflagplanedirection) * ê₃ + -dot(ê₄, κflagplanedirection) * ê₄)))\nglobal v = LinearAlgebra.normalize(vec((-dot(ê₃, ωflagplanedirection) * ê₃ + -dot(ê₄, ωflagplanedirection) * ê₄)))\np = 𝕍(LinearAlgebra.normalize(u + v))\nglobal p = -dot(ê₃, p) * ê₃ + -dot(ê₄, p) * ê₄\naxis = normalize(ℝ³(vec(p)[2:4]))\nM = mat4(Quaternion(progress * 4π, axis))\nο_transformed = M * Quaternion(vec(ο))\nι_transformed = M * Quaternion(vec(ι))","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproductspositivechina)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"For example, the Standard Model is formulated on 4-dimensional Minkowski spacetime, over which all fiber bundles can be trivialized and spinors have a simple explicit description. For the Symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In theoretical physics, Lie groups like the Lorentz and Poincaré groups, which are related to spacetime symmetries, and gauge groups, defining internal symmetries, are important cornerstones. Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group. The Lie algebra SL(2mathbbC) plays a special role in physics, because as a real Lie algebra it is isomorphic to the Lie algebra of the Lorentz group of 4-dimensional spacetime. At least locally, fields in physics can be described by maps on spacetime with values in vector spaces.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. As we will discuss in Sect. 6.8.2 in more detail, the group SL(2mathbbC) is the (orthochronous) Lorentz spin group, i.e. the universal covering of the identity component of the Lorentz group of 4-dimensional spacetime. The fundamental geometric opbject in a gauge theory is a principal bundle over spacetime with structure group given by the gauge group. The fibers of a principal bundle are sometimes thought of as an internal space at every spacetime point, not belonging to spacetime itself. Fiber bundles are indispensible in gauge theory and physics in the situation where spacetime, the base manifold, has a non-trivial topology.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It also happens if we compactify (Euclidean) spacetime mathbbR^4 to the 4-sphere S^4. In these situations, fields on spacetime often cannot be described simply by a map to a fixed vector space, but rather as sections of a non-trivial vector bundle. We will see that this is similar to the difference in special relativity between Minkowski spacetime and the choice of an inertial system. This can be compared, in special relativity, to the choice of an inertial system for Minkowski spacetime M, which defines an identification on M cong mathbbR^4. Of course, different choices of gauges are possible, leading to different trivializations of the principal bundle, just as different choices of inertial systems lead to different identifications of spacetime with mathbbR^4.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Note that, if we consider principal bundles over Minkowski spacetime mathbbR^4, it does not matter for this discussion that principal bundles over Euclidean spaces are always trivial by Corollary 4.2.9. This is very similar to special relativity, where spacetime is trivial, i.e. isomorphic to mathbbR^4 with a Minkowski metric, but what matters is the independence of the actual trivialization, i.e. the choice of inertial system. Table 4.2 Comparison between notions for special relativity and gauge theory","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":" Manifold Trivialization Transformations and invariance\nSpecial relativity Spacetime M M cong mathbbR^4 via inertial system Lorentz\nGauge theory Principal bundle P to M P cong M times G via choice of gauge Gauge","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It follows that, given a local gauge of the gauge bundle P, the section in E corresponds to a unique local map from spacetime into the vector space V. In particular, we can describe matter fields on a spacetime diffeomorphic to mathbbR^4 by unique maps from mathbbR^4 into a vector space, once a global gauge for the principal bundle has been chosen. At least locally (after a choice of local gauge) we can interpret connection 1-forms as fields on spacetime (the base manifold) with values in the Lie algebra of the gauge group. Notice that connections are not unique (if dim M dim G ge 1), not even in the case of trivial principal bundles (all connections that appear in the Standard Model over Minkowski spacetime, for example, are defined on trivial principal bundles). The diffeomorphism group Diff(M) of spacetime M plays a comparable role in general relativity.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"This is related to the fact that gauge theories describe local interactions (the interactions occur in single spacetime points). The local connection 1-form is thus defined on an open subset in the base manifold M and can be considered as a \"field on spacetime\" in the usual sense. Generalized Electric and Magnetic Fields on Minkowski Spacetime of Dimension 4 In quantum field theory, the gauge field A_mu is a function on spacetime with values in the operators on the Hilbert state space V (if we ignore for the moment questions of whether this operator is well-defined and issues of regularization). By Corollary 5.13.5 this difference can be identified with a 1-form on spacetime M with values in Ad(P).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In physics this fact is expressed by saying that gauge bosons, the differences A_mu-A_mu^0, are fields on spacetime that transform in the adjoint representation of G under gauge transformations. In the case of Minkowski spacetime, rotations correspond to Lorentz transformations. The pseudo-Riemannian case, like the case of Minkowski spacetime, is discussed less often, even though it is very important for physics (a notable exception is the thorough discussion in Helga Baun's book [13]). mathbbR^s1 and mathbbR^1t are the two versions of Minkowski spacetime (both versions are used in physics). This includes the particular case of the Lorentz group of Minkowski spacetime.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"However, as mentioned above, depending on the convention, 4-dimensional Minkowski spacetime in quantum field theory can have signature (+---), so that time carries the plus sign. Example 6.1.20 For applications concerning the Standard Model, the most important of these groups is the proper orthochronous Lorentz group SO^+(13) cong SO^+(31) of 4-dimensional Minkoeski spacetime. They are physical gamma matrices for Cl(13), i.e. for the Clifford algebra of Minkowski spacetime with signature (+---), in the so-called Weyl representation or chiral representation. Example 6.3.18 Let Gamma_a and gamma_a = i Gamma_a be the physical and mathematical gamma matrices for Cl(13) considered in Example 6.3.17. If we set Gamma_a^prime = gamma_a, gamma_a^prime = i Gamma_a^prime = -Gamma_a, then these are physical and Mathematical gamma matrices for Cl(13) of Minkowski spacetime with signature (-+++). Example 6.3.24 For Minkowski spacetime of dimension 4 we have Table 6.1 Complex Clifford algebras","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"n Cl(n) Cl^0(n) N\nEvan End(mathbbC^N) End(mathbbC^N2) oplus End(mathbbC^N2) 2^n2\nOdd End(mathbbC^N) oplus End(mathbbC^N) End(mathbbC^N) 2^(n-1)2","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Table 6.2 Real Clifford algebras","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"rho mod 8 Cl(st) N\n0 End(mathbbR^N) 2^n2\n1 End(mathbbC^N) 2^(n-1)2\n2 End(mathbbH^N) 2^(n-2)2\n3 End(mathbbH^N) oplus End(mathbbH^N) 2^(n-3)2\n4 End(mathbbH^N) 2^(n-2)2\n5 End(mathbbC^N) 2^(n-1)2\n6 End(mathbbR) 2^n2\n7 End(mathbbR^N) oplus End(mathbbR^N) 2^(n-1)2","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Table 6.3 Even part of real Clifford algebras","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"rho mod 8 Cl^0(st) N\n0 End(mathbbR^N) oplus End(mathbbR^N) 2^(n-2)2\n1 End(mathbbR^N) 2^(n-1)2\n2 End(mathbbC^N) 2^(n-2)2\n3 End(mathbbH^N) 2^(n-3)2\n4 End(mathbbH^N) oplus End(mathbbH^N) 2^(n-4)2\n5 End(mathbbH^N) 2^(n-3)2\n6 End(mathbbC^N) 2^(n-2)2\n7 End(mathbbR^N) 2^(n-1)2","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Cl(13) cong End(mathbbR^4)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Cl(31) cong End(mathbbH^2)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Cl^0(13) cong Cl^0(31) cong End(mathbbC^2)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Example 6.6.7 For Minkowski spacetime mathbbR^n-11 of dimension n we have n = rho + 2.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"We see that in Minkowski spacetime of dimension 4 there exist both Majorana and Weyl spinors of real dimension 4, but not Majorana-Weyl spinors. In quantum field theory, spinors become fields of operators on spacetime acting on a Hilbert space. Explicit formulas for Minkowski Spacetime of Dimension 4 We collect some explicit formulas concerning Clifford algebras and spinors for the case of 4-dimensional Minkowski spacetime. In Minkowski spacetime of dimension 4 and signature (+---) (usually used in quantum field theory) there exist both Weyl and Majorana spinors, but not Majorana-Weyl spinors. Our aim in this subsection is to prove that the orthochronous spin group Spin^+(13) of 4-dimensional Minkowski spacetime is isomorphic to the 6-dimensional Lie group SL(2mathbbC).","category":"page"},{"location":"newsreport.html#The-Story","page":"News Report","title":"The Story","text":"","category":"section"},{"location":"newsreport.html#Who","page":"News Report","title":"Who","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"With the discovery of a new particle, announced on 4 July 2012 at CERN, whose properties are \"consistent with the long-sought Higgs boson\" [31], the final elementary particle predicted by the classical Standard Model of particle physics has been found. Interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs bosons), determined by the Lagrangian. The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. The fact that there are 8 gluons, 3 weak gauge bosons, and 1 photon is related to the dimensions of the Lie groups SU(3) and SU(2) times U(1). Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields. The gauge bosons corresponding to these gauge groups are described by the adjoint representation that we discuss in Sect. 2.1.5. The representation Ad_H describes the representation of the gauge boson fields in the Standard Model. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Connections on principal bundles, that we discuss in Chap. 5, correspond to gauge fields whose particle excitations in the associated quantum field theory are the gauge bosons that transmit interactions. These fields are often called gauge fields and correspond in the associated quatum field theory to gauge bosons. This implies a direct interaction between gauge bosons (the gluons in QCD) that does not occur in abelian gauge theories like quantum electrodynamics (QED). The difficulties that are still present nowadays in trying to understand the quantum version of non-abelian gauge theories, like quantum chromodynamics, can ultimately be traced back to this interaction between gauge bosons. The real-valued fields A_mu^a in C^infty(UmathbbR) and the corresponding real-valued 1-forms A_s in Omega^1(U) are called (local) gauge boson fields.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In physics, the quadratic term A_mu A_nu in the expression for F_munu (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field A_mu. This explains why gluons, the gauge bosons of QCD, interact directly with each other, while photons, the gauge bosons of QED, do not. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by A_mu and the particles described by the field phi. We then get a better understanding of why gauge bosons in physics are said to transform under the adjoint representation.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Strictly speaking, gauge bosons, the excitations of the gauge field, should then be described classically by the difference A - A^0, where A is some other connection 1-form and not by the field A itself. In physics this fact is expressed by saying that gauge bosons, the differences A_mu - A_mu^0, are fields on spacetime that transform in the adjoint represntation of G under gauge transformations. Gauge fields correspond to gauge bosons (spin 1 particles) and are described by 1-forms or, dually, vector fields. Even though spinors are elementary objects, some of their properties (like the periodicity modulo 8, real and quaternionic structures, or bilinear and Hamiltonian scalar products) are not at all obvious, already on the level of linear algebra, and do not have a direct analogue in the bosonic world of vectors and tensors. The existence of gauge symmetries is particularly important: it can be shown that a quantum field theory involving massless spin 1 bosons can be consistent (i.e. unitary, see Sect. 7.1.3) only if it is gauge invariant [125,143].","category":"page"},{"location":"newsreport.html#Graph","page":"News Report","title":"Graph","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: graph)","category":"page"},{"location":"newsreport.html#What","page":"News Report","title":"What","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. Spin groups such as the universal covering of the Lorentz group and its higher dimesnional analogues, are also important in physics, because they are involved in the mathematical description of fermions. Counting in this way, the Standard Model thus contains at the most elementary level 90 fermions (particles and antiparticles). The complex vector space V of fermions, which carries a representation of G, has dimension 45 (plus the same number of corresponding antiparticles) and is the direct sum of the two G-invariant subspaces (sectors): a lepton sector of dimension 9 (where we do not include the hypothetical right-handed neutrinos) and a quark sector of dimension 36. Matter fields in the Standard Model, like quarks and leptons, or sacalar fields, like the Higgs field, correspond to sections of vector bundles associated to the principal bundle (and twisted by spinor bundles in the case of fermions).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"For example, in the Standard Model, one generation of fermions is described by associated complex vector bundles of rank 8 for left-handed fermions and rank 7 for right-handed fermions, associated to representations of the gauge group SU(3) times SU(2) times U(1). Matter fields in physics are described by smooth sections of vector bundles E associated to principal bundles P via the representations of the gauge group G on a vector space V (in the case of fermions the associated bundle E is twisted in addition with a spinor bundle S, i.e. the bundle is S otimes E). Additional matter fields, like fermions or scalars, can be introduced using associated vector bundles. These particles are fermions (spin frac12 particles) and are described by spinor fields (spinors). Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly neutrinos) and, together with the Dirac operator, the kinetic term and the interaction term; see Sect. 7.6. This is related to the fact that the weak interaction in the Standard Model is not invariant under parity inversion that exchanges left-handed with right-handed fermions.","category":"page"},{"location":"newsreport.html#Perspective","page":"News Report","title":"Perspective","text":"","category":"section"},{"location":"newsreport.html#How","page":"News Report","title":"How","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Hence, by the uniqueness of integral curves (which is a theorem about the uniqueness of solutions to odrinary differential equations) we have phi_X(s) cdot phi_X(t) = phi_X(s + t) forall t in I cap (t_min - s t_max - s). This implies the claim by uniqueness of solutions of ordinary differential equations. The unique solution of this differential equation for gamma(t) is gamma(t) = e^tr(X)t. Then e^D = beginbmatrix e^d_1 0 0 0 e^d_2 0 ddots 0 0 e^d_n endbmatrix and the equation det(e^D) = e^d_1 e^d_n = e^d_1 + + d_n = e^tr(D) is trivially satisfied. Then we can calculate: (R^*_gs)_p(XY) = L_(pg)^-1*R_g*(X) L_(pg)^-1*R_g*(Y) = Ad_g^-1 circ L_p^-1*(X) Ad_g^-1 circ L_p^-1*(Y) and s_p(XY) = L_p^-1*(X) L_p^-1*(Y), where in both equations we used that s is left invariant.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Lemma 3.3.3 For A in Mat(m times m mathbbH) and v in mathbbH^m the following equation holds: detbeginbmatrix1 v 0 Aendbmatrix = det(A). Lemma 4.1.13 (Cocycle Conditions) The transition functions phi_ij_ij in I satisfy the following equations:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"phi_ii(x) = Id_F for x in U_i,\nphi_ij(x) circ phi_ji(x) = Id_F for x in U_i cap U_j,\nphi_ik(x) circ phi_kj(x) circ phi_ji(x) = Id_F for x in U_i cap U_j cap U_k.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The third equation is called the cycycle condition.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"5.5.2 The structure equation Theorem 5.5.4 (Structure Equation) The curvature form F of a connection form A satisfies F = dA + frac12AA.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proof We check the formula by inserting XY in T_pP on both sides of the equation, where we distinguish the following three cases:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Both X and Y are vertical: Then X and Y are fundamental vectors, X = tildeV_p Y = tildeW_p for certain elements VW in g. We get F(XY) = dA(pi_H(X) pi_H(Y)) = 0. On the other hand we have frac12AA(XY) = A(X)A(Y) = VW. The differential dA of a 1-form A is given according to Proposition A.2.22 by dA(XY) = L_X(A(Y))-L_Y(A(X))-A(XY), where we extend the vectors X and Y to vector fields in a neighbourhood of p. If we choose the extension by fundamental vector fields tildeV and tildeW, then dA(XY) = L_X(W) - L_Y(V) - VW = -VW since V and W are constant maps from P to g and we used that tildeVtildeW = tildeVW according to Proposition 3.4.4. This implies the claim.\nBoth X and Y are horizontal: Then F(XY) = dA(XY) and frac12AA(XY) = A(X) A(Y) = 00=0. This implies the claim.\nX is vertical and Y is horizontal: Then X = tildeV_p for some V in g. We have F(XY) = dA(pi_H(X)pi_H(Y)) = dA(0 Y) = 0 and frac12AA(XY) = A(X)A(Y) - V0 = 0. Furthermore, dA(XY) = L_tildeV(A(Y)) - L_Y(V) - A(tildeVY) = -A(tildeVY) = 0 since tildeVY is horizontal by Lemma 5.5.5. This implies the claim.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The structure equation is very useful when we want to calculate the curvature of a given connection.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"By the structure equation we have F = dA + frac12 A A so that dF = frac12 dA A.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proposition 5.6.2 (Local Structure Equation) The local field strength can be calculated as F_s = dA_s + frac12A_sA_s and F_munu = partial_mu A_nu - partial_nu A_mu + A_mu A_nu.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It remains to check that F_M is closed. In a local gauge s we have according to the local structure equation F_s = dA_s + frac12A_sA_s.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proposition 5.6.8 For the connection on the Hopf bundle the following equation holds: frac12pi i int_S^2 F_S^2 = 1.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"We write A_mu = A_s(partial_mu) F_munu = F_s(partial_mu partial_nu) and we have the local structure equation F_munu = partial_mu A_nu - partial_nu A_mu + A_mu A_nu.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"We will determine g(t) as the solution of a differential equation.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proof Properties 1-3 follow from the theory of ordinary differential equations. (Parallel transport)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Recall that for the proof of Theorem 5.8.2 concerning the existence of a horizontal lift gamma^* of a curve gamma01 to M where gamma^*(0) = p in P_gamma(0), we had to solve the differential equation dotg(t) = -R_g(t)* A(dotdelta(t)), with g(0) = e, where delta is some lift of gamma and g01 to G is a map with gamma^*(t) = delta(t) cdot g(t).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Then the differential equation can be written as fracdg(t)dt = -A_s(dotgamma(t)) cdot g(t).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Path-ordered exponentials are useful, because they define solutions to the ordinary differential equation we are interested in.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Then uniqueness of the solution to ordinary differential equations show that g equiv h, hence g takes values in G.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The solution to this differential equation is","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"g(t) = P exp(- int_0^t sum_mu=1^n A_smu(gamma(s))fracdx^mudsds) = P exp(- int_gamma(0)^gamma(t) sum_mu=1^n A_smu (x^mu) dx^mu) = P exp(- int_gamma_t A_s), where gamma_t denotes the restriction of the curve gamma to 0t.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"What is the interpretation of the structure equation?","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Taking the determinant of both sides of this equation shows that:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Lemma 6.1.7 Matrices A in O(st) satisfy detA = pm 1.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"A^T beginbmatrix I_s 0 0 -I_t endbmatrix A = beginbmatrix I_s 0 0 -I_t endbmatrix.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Remark 6.2.5 We can think of the linear map gamma as a linear square root of the symmetric bilinear form -Q: in the definition of Clifford algebras, it suffices to demand that gamma(v)^2 = -Q(vv) cdot 1 forall v w in V, because, considering this equation for vectors v w v + w, the equation gamma(v) gamma(w = -2Q(v w) cdot 1 forall v w in V follows.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Lemma 6.3.6 Every chirality element omega satisfies","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"omegae_a = 0\nomegae_a cdot e_b = 0 forall 1 le a b le n.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proof The first equation follows from e_a cdot omega = lambda e_a cdot e_1 e_n = (-1)^a - 1 lambda e_1 e_a cdot e_a e_n","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"omega cdot e_a = lambda e_1 e_n cdot e_a = (-1)^n - a lambda e_1 e_a cdot e_a e_n = -e_a cdot omega,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"since n is even. The second equation is a consequence of the first.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Let Gamma_1 Gamma_n be physical gamma matrices. We set","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma_a = eta^ac Gamma_c,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^bc = frac12 Gamma^b Gamma^c = frac12 (Gamma^b Gamma^c - Gamma^c Gamma^b),","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^n + 1 = -i^k + t Gamma^1 Gamma^n","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"and similarly for the mathematical gamma-matrices (in the first equation there is an implicit sum over c; this is an instance of the Einstein summation convention). These matrices satisfy by Lemma 6.3.6","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^n + 1 Gamma^a = 0,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^n + 1 Gamma^bc = 0,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"gamma^bc = -Gamma^bc.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In the following examples we use the Pauli matrices","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"sigma_1 = beginbmatrix 0 1 1 0 endbmatrix, sigma_2 = beginbmatrix 0 -i i 0 endbmatrix, sigma_3 = beginbmatrix 1 0 0 -1 endbmatrix.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It is easy to check that they satisfy the identities","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"sigma^2 = I_2 j = 1 2 3,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"sigma_j sigma_j + 1 = -sigma_j + 1 sigma_j = i sigma_j + 2 j = 1 2 3,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"where in the second equation j + 1 and j + 2 are taken mod 3.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(psi phi) = psi^T C phi","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Furthermore property 1. and 2. in Definition 6.7.1 are equivalent to","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"gamma_a^T = mu C gamma_a C^-1 for all a = 1 s + t.\nC^T = nu C.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The first equation also holds with the physical Clifford matrices Gamma_a instead of the mathematical matrices gamma_a.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"There is an equivalent equation to the first one with physical Clifford matrices Gamma_a 1 cdot Gamma^dagger_a = -delta A Gamma_a A^-1 for all a = 1 s + t.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Furthermore, property 1. and 2. in Definition 6.7.8 are equivalent to:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"gamma_a^dagger = delta A gamma_a A^-1 for all a = 1 s + t.\nA^dagger = A.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Given a spin structure on a pseudo-Riemannian manifold and the spinor bundle S, we would like to have a covariant derivative on S so that we can define field equations involving derivatives of spinors.","category":"page"},{"location":"newsreport.html#The-Iconic-Wall","page":"News Report","title":"The Iconic Wall","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: corrected-wall)","category":"page"},{"location":"newsreport.html#Tome","page":"News Report","title":"Tome","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: tome)","category":"page"},{"location":"newsreport.html#Wrap-Up","page":"News Report","title":"Wrap Up","text":"","category":"section"},{"location":"newsreport.html#Why","page":"News Report","title":"Why","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The following three chapters discuss applications in physics: the Lagrangians and interactions in the Standard Model, spontaneous symmetry breaking, the Higgs mechanism of mass generation, and some more advanced and modern topics like neutrino masses and CP violation. Depending on the time, the interests and the prior knowledge of the reader, he or she can take a shortcut and immediately start at the chapters on connections, spinors or Lagrangians, and then go back if more detailed mathematical knowledge is required at some point. An interesting and perhaps underappreciated fact is that a substantial number of phenomena in particle physics can be understood by analysing representations of Lie groups and by rewriting or rearranging Lagrangians.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Symmetries of Lagrangians interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs boson), determined by the Lagrangian. For the symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In the following chapter we will study some associated concepts, like representations (which are used to define the actions of Lie groups on fields) and invariant matrices (which are important in the construction of the gauge invariant Yang-Mills Lagrangian). We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The existence of positive definite Ad-invariant scalar products on the Lie algebra of compact Lie groups is very important in gauge theory, in particular, for the construction of the gauge-invariant Yang-Mills Lagrangian; see Sect. 7.3.1. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]). In a gauge-invariant Lagrangian this results in terms of order higher than two in the matter and gauge fields, which are interpreted as interactions between the corresponding particles. In non-abelian gauge theories, like quantum chromodynamics (QCD), there are also terms in the Lagrangian of order higher than two in the gauge fields themselves, coming from a quadratic term in the curvature that appears in the Yang-Mills Lagrangian.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In physics, the quadratic term A_mu A_nu in the expression for F_munu (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field A_mu. These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by A_mu and the particles described by the field phi.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: feynmandiagrams)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Figure 5.2 shows the Feynman diagrams for the cubic and quartic terms which appear in the Klein-Gordon Lagrangian in Eq. (7.3), representing the interaction between a gauge field A and a charged scalar field described locally by a map phi with values in V. Fig 5.2 Feynman diagrams for interaction between gauge field and charged scalar","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Hermitian scalar products are particularly important, because we need them in Chap. 7 to define Lorentz invariant Lagrangians involving spinors.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"psi phi = overlinepsi phi,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"overlinepsi = psi^dagger A.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly the neutrinos) and, together with the Dirac operator the kinetic term and the interaction term; see Sect. 7.6.","category":"page"},{"location":"newsreport.html#Porta.jl","page":"News Report","title":"Porta.jl","text":"","category":"section"},{"location":"newsreport.html#References","page":"News Report","title":"References","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Mark J.D. Hamilton, Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics, Springer Cham, DOI, published: 10 January 2018.\nSir Roger Penrose, The Road to Reality, (2004).\nRoger Penrose, Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).\nRichard M. Murray and Zexiang Li, A Mathematical Introduction to Robotic Manipulation, 1st Edition, 1994, CRC Press, read, buy.\nEdward Witten, Physics and Geometry, (1987).\nThe iconic Wall of Stony Brook University.","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Description = \"How the reaction wheel unicycle works.\"","category":"page"},{"location":"reactionwheelunicycle.html#The-Reaction-Wheel-Unicycle","page":"Reaction Wheel Unicycle","title":"The Reaction Wheel Unicycle","text":"","category":"section"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_cnt = beginbmatrix dotx - r_w dottheta cos(delta) newline doty - r_w dottheta sin(delta) newline dotz endbmatrix = beginbmatrix 0 newline 0 newline 0 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"dotx = r_w dottheta cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"doty = r_w dottheta sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"dotz = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotq_i) - fracpartial Lpartial q_i = Q_i + sum_k=1^n lambda_k a_ki","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"i = 1 ldots m","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"L = T_total - P_total","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_w2^cpT = beginbmatrix 1 0 0 0 newline 0 cos(alpha) -sin(alpha) 0 newline 0 sin(alpha) cos(alpha) 0 newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 r_w newline 0 0 0 1 endbmatrix = beginbmatrix 1 0 0 0 newline 0 cos(alpha) -sin(alpha) -r_w sin(alpha) newline 0 sin(alpha) cos(alpha) r_w cos(alpha) newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_cp^gT = beginbmatrix cos(delta) -sin(delta) 0 x newline sin(delta) cos(delta) 0 y newline 0 0 1 0 newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_w2^gT = _cp^gT times _w2^cpT = beginbmatrix cos(delta) -sin(delta) cos(alpha) sin(delta) sin(alpha) x + r_w sin(delta) sin(alpha) newline sin(delta) cos(delta) cos(alpha) -cos(delta) sin(alpha) y - r_w cos(delta) sin(alpha) newline 0 sin(alpha) cos(alpha) r_w cos(alpha) newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^w2P_w = beginbmatrix 0 newline 0 newline 0 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gP_w = _w2^gT times ^w2P_w = beginbmatrix x + r_w sin(alpha) sin(delta) newline y - r_w sin(alpha) cos(delta) newline r_w cos(alpha) newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^w2T = beginbmatrix cos(beta) 0 sin(beta) 0 newline 0 1 0 0 newline -sin(beta) 0 cos(beta) 0 newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 l_c newline 0 0 0 1 endbmatrix = beginbmatrix cos(beta) 0 sin(beta) l_c sin(beta) newline 0 1 0 0 newline -sin(beta) 0 cos(beta) l_c cos(beta) newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gT = _w2^gT times _c^w2T = beginbmatrix _c^gt_11 -sin(delta) cos(alpha) _c^gt_13 _c^gt_14 newline _c^gt_21 cos(delta) cos(alpha) _c^gt_23 _c^gt_24 newline -cos(alpha) sin(beta) sin(alpha) cos(alpha) cos(beta) _c^gt_34 newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_11 = cos(beta) cos(delta) - sin(alpha) sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_13 = sin(beta) cos(delta) + sin(alpha) cos(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_14 = x + r_w sin(delta) sin(alpha) + l_c sin(beta) cos(delta) + l_c sin(alpha) cos(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_21 = cos(beta) sin(delta) + sin(alpha) sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_23 = sin(beta) sin(delta) - sin(alpha) cos(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_24 = y - r_w cos(delta) sin(alpha) + l_c sin(beta) sin(delta) - l_c sin(alpha) cos(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_34 = r_w cos(alpha) + l_c cos(alpha) cos(beta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^cP_c = beginbmatrix 0 newline 0 newline 0 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gP_c = _c^gT times ^cP_c = beginbmatrix ^gp_c1 newline ^gp_c2 newline ^gp_c3 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_c1 = x + r_w sin(alpha) sin(delta) + l_c cos(beta) sin(alpha) sin(delta) + l_c sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_c2 = y - r_w sin(alpha) cos(delta) - l_c cos(beta) sin(alpha) cos(delta) + l_c sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_c3 = r_w cos(alpha) + l_c cos(beta) cos(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^cT = beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 l_cr newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 cos(gamma) -sin(gamma) 0 newline 0 sin(gamma) cos(gamma) 0 newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 0 newline 0 0 0 1 endbmatrix = beginbmatrix 1 0 0 0 newline 0 cos(gamma) -sin(gamma) 0 newline 0 sin(gamma) cos(gamma) l_cr newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gT = _c^gT times _r^cT = beginbmatrix _r^gt_11 _r^gt_12 _r^gt_13 _r^gt_14 newline _r^gt_21 _r^gt_22 _r^gt_23 _r^gt_24 newline -cos(alpha) sin(beta) _r^gt_32 _r^gt_33 _r^gt_34 newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_11 = cos(beta) cos(delta) - sin(alpha) sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_12 = -sin(delta) cos(alpha) cos(gamma) + cos(delta) sin(beta) sin(gamma) + sin(delta) sin(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_13 = sin(delta) cos(alpha) sin(gamma) + cos(delta) sin(beta) cos(gamma) + sin(delta) sin(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_14 = 0 + l_cr (cos(delta) sin(beta) + sin(delta) sin(alpha) cos(beta)) + l_c sin(beta) cos(delta) + l_c cos(beta) sin(delta) sin(alpha) + x + r_w sin(delta) sin(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_21 = cos(beta) sin(delta) + sin(alpha) sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_22 = cos(delta) cos(alpha) cos(gamma) + sin(delta) sin(beta) sin(gamma) - cos(delta) sin(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_23 = -cos(delta) cos(alpha) sin(gamma) + sin(delta) sin(beta) cos(gamma) - cos(delta) sin(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_24 = l_cr (sin(delta) sin(beta) - cos(delta) sin(alpha) cos(beta)) + l_c sin(beta) sin(delta) - l_c cos(beta) cos(delta) sin(alpha) + y - r_w cos(delta) sin(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_32 = sin(alpha) cos(gamma) + cos(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_33 = -sin(alpha) sin(gamma) + cos(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_34 = l_cr cos(alpha) cos(beta) + l_c cos(beta) cos(alpha) + r_w cos(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^rP_r = beginbmatrix 0 newline 0 newline 0 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gP_r = _r^gT times ^rP_r = beginbmatrix ^gp_r1 newline ^gp_r2 newline ^gp_r3 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_r1 = x + r_w sin(alpha) sin(delta) + (l_c + l_cr) cos(beta) sin(alpha) sin(delta) + (l_c + l_cr) sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_r2 = y - r_w sin(alpha) cos(delta) - (l_c + l_cr) cos(beta) sin(alpha) cos(delta) + (l_c + l_cr) sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_r3 = r_w cos(alpha) + (l_c + l_cr) cos(beta) cos(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_w = fracdP_wdt","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_c = fracdP_cdt","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_r = fracdP_rdt","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Omega_w = beginbmatrix 0 newline dottheta newline 0 newline 0 endbmatrix + beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _g^w2T times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix 0 newline dottheta newline 0 newline 0 endbmatrix + beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _w2^gT^-1 times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotalpha newline dottheta + dotdelta sin(alpha) newline dotdelta cos(alpha) endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Omega_c = beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _w2^cT times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _g^cT times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _c^w2T^-1 times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _c^gT^-1 times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotalpha cos(beta) - dotdelta cos(alpha) sin(beta) newline dotbeta + dotdelta sin(alpha) newline dotalpha sin(beta) + dotdelta cos(alpha) cos(beta) newline 0 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^w2T = _w2^gT^-1 times _r^gT","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Omega_r = beginbmatrix dotgamma newline 0 newline 0 newline 0 endbmatrix + _c^rT times beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _w2^rT times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _g^rT times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotgamma newline 0 newline 0 newline 0 endbmatrix + _r^cT^-1 times beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _r^w2T^-1 times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _r^gT^-1 times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotgamma + dotalpha cos(beta) - dotdelta cos(alpha) sin(beta) newline omega_r2 newline omega_r3 newline 0 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"omega_r2 = dotbeta cos(gamma) + dotalpha sin(beta) sin(gamma) + dotdelta sin(alpha) cos(gamma) + dotdelta cos(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"omega_r3 = -dotbeta sin(gamma) + dotalpha sin(beta) cos(gamma) - dotdelta sin(alpha) sin(gamma) + dotdelta cos(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_w = frac12 m_w V_w^T V_w + frac12 Omega_w^T I_w Omega_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_w = m_w g P_w(3)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_c = frac12 m_c V_c^T V_c + frac12 Omega_c^T I_c Omega_c","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_c = m_c g P_c(3)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_r = frac12 m_r V_r^T V_r + frac12 Omega_r^T I_r Omega_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_r = m_r g P_r(3)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_total = T_w + T_c + T_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_total = P_w + P_c + P_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m = 7 n = 2","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotx) - fracpartial Lpartial x = lambda_1","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial doty) - fracpartial Lpartial y = lambda_2","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dottheta) - fracpartial Lpartial theta = tau_w - r_w cos(delta) lambda_1 - r_w sin(delta) lambda_2","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotbeta) - fracpartial Lpartial beta = -tau_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotalpha) - fracpartial Lpartial alpha = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotgamma) - fracpartial Lpartial gamma = tau_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotdelta) - fracpartial Lpartial delta = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Wheel dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_11 ddotbeta + m_12 ddotgamma + m_13 ddotdelta + m_14 ddottheta + c_11 dotbeta^2 + c_12 dotgamma^2 + c_13 dotdelta^2 + c_14 dotalpha dotdelta + c_15 dotbeta dotgamma + c_16 dotbeta dotdelta + c_17 dotgamma dotdelta = tau_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Chassis longitudinal dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_21 ddotalpha + m_22 ddotbeta + m_23 ddotdelta + m_24 ddottheta + c_21 dotalpha^2 + c_22 dotdelta^2 + c_23 dotalpha dotgamma + c_24 dotalpha dotdelta + c_25 dotbeta dotgamma + c_26 dotgamma dotdelta + c_27 dotdelta dottheta + g_21 = -tau_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Chassis lateral dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_31 ddotalpha + m_32 ddotbeta + m_33 ddotgamma + m_34 ddotdelta + c_31 dotbeta^2 + c_32 dotgamma^2 + c_33 dotdelta^2 + c_34 dotalpha dotbeta + c_35 dotalpha dotgamma + c_36 dotbeta dotgamma + c_37 dotbeta dotdelta + c_38 dotgamma dotdelta + c_39 dotdelta dottheta = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Reaction wheel dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_41 ddotalpha + m_42 ddotgamma + m_43 ddotdelta + m_44 ddottheta + c_41 dotalpha^2 + c_42 dotbeta^2 + c_43 dotdelta^2 + c_44 dotalpha dotbeta + c_45 dotalpha dotdelta + c_46 dotbeta dotdelta + c_47 dotdelta dottheta + g_41 = tau_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Turning dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_51 ddotalpha + m_52 ddotbeta + m_53 ddotgamma + m_54 ddotdelta + m_55 ddottheta + c_51 dotalpha^2 + c_52 dotbeta^2 + c_53 dotgamma^2 + c_54 dotalpha dotbeta + c_55 dotalpha dotgamma + c_56 dotalpha dotdelta + c_57 dotalpha dottheta + c_58 dotbeta dotgamma + c_59 dotbeta dotdelta + c_510 dotgamma dotdelta + c_511 dotdelta dottheta = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracmathrmd xleft( t right)mathrmdt = r_w cosleft( deltaleft( t right) right) fracmathrmd thetaleft( t right)mathrmdt newline fracmathrmd yleft( t right)mathrmdt = r_w sinleft( deltaleft( t right) right) fracmathrmd thetaleft( t right)mathrmdt newline fracmathrmd zleft( t right)mathrmdt = 0 newline I_w = left beginarraycccc I_w1 0 0 0 newline 0 I_w2 0 0 newline 0 0 I_w3 0 newline 0 0 0 0 newline endarray right newline I_c = left beginarraycccc I_c1 0 0 0 newline 0 I_c2 0 0 newline 0 0 I_c3 0 newline 0 0 0 0 newline endarray right newline I_r = left beginarraycccc I_r1 0 0 0 newline 0 I_r2 0 0 newline 0 0 I_r3 0 newline 0 0 0 0 newline endarray right newline mathrmw2cpTleft( t right) = left beginarraycccc 1 0 0 0 newline 0 cosleft( alphaleft( t right) right) - sinleft( alphaleft( t right) right) - r_w sinleft( alphaleft( t right) right) newline 0 sinleft( alphaleft( t right) right) cosleft( alphaleft( t right) right) r_w cosleft( alphaleft( t right) right) newline 0 0 0 1 newline endarray right newline mathrmcpgTleft( t right) = left beginarraycccc cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) 0 xleft( t right) newline sinleft( deltaleft( t right) right) cosleft( deltaleft( t right) right) 0 yleft( t right) newline 0 0 1 0 newline 0 0 0 1 newline endarray right newline mathrmw2gTleft( t right) = mathrmcpgTleft( t right) mathrmw2cpTleft( t right) newline w2P_w = left beginarrayc 0 newline 0 newline 0 newline 1 newline endarray right newline mathrmgP_wleft( t right) = mathrmw2gTleft( t right) w2P_w newline mathrmcw2Tleft( t right) = left beginarraycccc cosleft( betaleft( t right) right) 0 sinleft( betaleft( t right) right) l_c sinleft( betaleft( t right) right) newline 0 1 0 0 newline -sinleft( betaleft( t right) right) 0 cosleft( betaleft( t right) right) l_c cosleft( betaleft( t right) right) newline 0 0 0 1 newline endarray right newline mathrmcgTleft( t right) = mathrmw2gTleft( t right) mathrmcw2Tleft( t right) newline cP_c = left beginarrayc 0 newline 0 newline 0 newline 1 newline endarray right newline mathrmgP_cleft( t right) = mathrmcgTleft( t right) cP_c newline mathrmrcTleft( t right) = left beginarraycccc 1 0 0 0 newline 0 cosleft( gammaleft( t right) right) - sinleft( gammaleft( t right) right) 0 newline 0 sinleft( gammaleft( t right) right) cosleft( gammaleft( t right) right) l_cr newline 0 0 0 1 newline endarray right newline mathrmrgTleft( t right) = mathrmcgTleft( t right) mathrmrcTleft( t right) newline rP_r = left beginarrayc 0 newline 0 newline 0 newline 1 newline endarray right newline mathrmgP_rleft( t right) = mathrmrgTleft( t right) rP_r newline mathrmrw2Tleft( t right) = mathrminvleft( mathrmw2gTleft( t right) right) mathrmrgTleft( t right) newline V_wleft( t right) = mathrmbroadcastleft( D mathrmgP_wleft( t right) right) newline V_cleft( t right) = mathrmbroadcastleft( D mathrmgP_cleft( t right) right) newline V_rleft( t right) = mathrmbroadcastleft( D mathrmgP_rleft( t right) right) newline Omega_wleft( t right) = mathrmbroadcastleft( + left beginarrayc _derivativeleft( alphaleft( t right) t 1 right) newline _derivativeleft( thetaleft( t right) t 1 right) newline 0 newline 0 newline endarray right mathrminvleft( mathrmw2gTleft( t right) right) left beginarrayc 0 newline 0 newline _derivativeleft( deltaleft( t right) t 1 right) newline 0 newline endarray right right) newline Omega_cleft( t right) = mathrmbroadcastleft( + mathrmbroadcastleft( + left beginarrayc 0 newline _derivativeleft( betaleft( t right) t 1 right) newline 0 newline 0 newline endarray right mathrminvleft( mathrmcw2Tleft( t right) right) left beginarrayc _derivativeleft( alphaleft( t right) t 1 right) newline 0 newline 0 newline 0 newline endarray right right) mathrminvleft( mathrmcgTleft( t right) right) left beginarrayc 0 newline 0 newline _derivativeleft( deltaleft( t right) t 1 right) newline 0 newline endarray right right) newline Omega_rleft( t right) = mathrmbroadcastleft( + mathrmbroadcastleft( + mathrmbroadcastleft( + left beginarrayc _derivativeleft( gammaleft( t right) t 1 right) newline 0 newline 0 newline 0 newline endarray right mathrminvleft( mathrmrcTleft( t right) right) left beginarrayc 0 newline _derivativeleft( betaleft( t right) t 1 right) newline 0 newline 0 newline endarray right right) mathrminvleft( mathrmrw2Tleft( t right) right) left beginarrayc _derivativeleft( alphaleft( t right) t 1 right) newline 0 newline 0 newline 0 newline endarray right right) mathrminvleft( mathrmrgTleft( t right) right) left beginarrayc 0 newline 0 newline _derivativeleft( deltaleft( t right) t 1 right) newline 0 newline endarray right right) newline T_wleft( t right) = mathrmadjointleft( V_wleft( t right) right) mathrmbroadcastleft( * V_wleft( t right) mathrmRefleft( 05 m_w right) right)_1 + mathrmadjointleft( Omega_wleft( t right) right) mathrmbroadcastleft( * I_w Omega_wleft( t right) 05 right)_1 newline P_wleft( t right) = g mathrmgP_wleft( t right)_3 m_w newline T_cleft( t right) = mathrmadjointleft( V_cleft( t right) right) mathrmbroadcastleft( * V_cleft( t right) mathrmRefleft( 05 m_c right) right)_1 + mathrmadjointleft( Omega_cleft( t right) right) mathrmbroadcastleft( * I_c Omega_cleft( t right) 05 right)_1 newline P_cleft( t right) = g mathrmgP_cleft( t right)_3 m_c newline T_rleft( t right) = mathrmadjointleft( V_rleft( t right) right) mathrmbroadcastleft( * V_rleft( t right) mathrmRefleft( 05 m_r right) right)_1 + mathrmadjointleft( Omega_rleft( t right) right) mathrmbroadcastleft( * I_r Omega_rleft( t right) 05 right)_1 newline P_rleft( t right) = g mathrmgP_rleft( t right)_3 m_r newline T_totalleft( t right) = T_rleft( t right) + T_cleft( t right) + T_wleft( t right) newline P_totalleft( t right) = P_wleft( t right) + P_cleft( t right) + P_rleft( t right) newline Lleft( t right) = T_totalleft( t right) - P_totalleft( t right) newline","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"L = 05 left( left( fracfracmathrmd alphaleft( t right)mathrmdt cosleft( betaleft( t right) right)sin^2left( betaleft( t right) right) + cos^2left( betaleft( t right) right) + fracleft( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + left( - sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) fracmathrmd deltaleft( t right)mathrmdtsinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( betaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) right)^2 I_c1 + left( fracmathrmd betaleft( t right)mathrmdt + fracleft( - left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - left( - sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtsinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( betaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) right)^2 I_c2 + left( fracsinleft( betaleft( t right) right) fracmathrmd alphaleft( t right)mathrmdtsin^2left( betaleft( t right) right) + cos^2left( betaleft( t right) right) + fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) fracmathrmd deltaleft( t right)mathrmdtsinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( betaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) right)^2 I_c3 right) + 05 left( left( fracmathrmd gammaleft( t right)mathrmdt + fracleft( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) fracmathrmd alphaleft( t right)mathrmdtleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( frac - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + fracleft( left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) sinleft( betaleft( t right) right) right) right)^2 I_r1 + left( fracleft( - left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) - left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) fracmathrmd alphaleft( t right)mathrmdtleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( frac - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + fracleft( - left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) sinleft( betaleft( t right) right) right) + fracfracmathrmd betaleft( t right)mathrmdt cosleft( gammaleft( t right) right)sin^2left( gammaleft( t right) right) + cos^2left( gammaleft( t right) right) right)^2 I_r2 + left( fracleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) fracmathrmd alphaleft( t right)mathrmdtleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( frac - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + fracleft( left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) sinleft( betaleft( t right) right) right) + frac - fracmathrmd betaleft( t right)mathrmdt sinleft( gammaleft( t right) right)sin^2left( gammaleft( t right) right) + cos^2left( gammaleft( t right) right) right)^2 I_r3 right) + 05 left( fracleft( fracmathrmd deltaleft( t right)mathrmdt right)^2 left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right)^2 I_w3left( cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right)^2 + left( fracmathrmd alphaleft( t right)mathrmdt right)^2 I_w1 + left( fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) fracmathrmd deltaleft( t right)mathrmdtcos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracmathrmd thetaleft( t right)mathrmdt right)^2 I_w2 right) + 05 m_c left( left( fracmathrmdmathrmdt left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt left( xleft( t right) + l_c sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + r_w sinleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) + l_c sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt 1 right)^2 + left( fracmathrmdmathrmdt left( yleft( t right) + l_c sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - r_w cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) - l_c cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 right) + 05 m_r left( left( fracmathrmdmathrmdt left( xleft( t right) + l_c sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + r_w sinleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) + l_c sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) + l_cr left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right)^2 + left( fracmathrmdmathrmdt left( yleft( t right) + l_c sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - r_w cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) - l_c cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) + l_cr left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right)^2 + left( fracmathrmdmathrmdt 1 right)^2 + left( fracmathrmdmathrmdt left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) + l_cr cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) right)^2 right) + 05 m_w left( left( fracmathrmdmathrmdt r_w cosleft( alphaleft( t right) right) right)^2 + left( fracmathrmdmathrmdt left( yleft( t right) - r_w cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt left( xleft( t right) + r_w sinleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt 1 right)^2 right) - g m_w r_w cosleft( alphaleft( t right) right) - g left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) m_c - g left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) + l_cr cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) m_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"left beginarrayc _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline endarray right = left beginarrayc lambda_1 newline lambda_2 newline tau_w - r_w sinleft( deltaleft( t right) right) lambda_2 - r_w cosleft( deltaleft( t right) right) lambda_1 newline -tau_w newline 0 newline tau_p newline 0 newline endarray right","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"Description = \"Read the documentation of the Porta.jl project.\"","category":"page"},{"location":"index.html#Geometrize-the-quantum!","page":"Home","title":"Geometrize the quantum!","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"This project is inspired by Eric Weinstein's Graph-Wall-Tome (GWT) project. Watch visual models on the YouTube channel.","category":"page"},{"location":"index.html#Requirements","page":"Home","title":"Requirements","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"CSV v0.10.13\nDataFrames v1.6.1\nFileIO v1.16.3\nGLMakie v0.9.9","category":"page"},{"location":"index.html#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"You can install Porta by running this (in the REPL):","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"]add Porta","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"or,","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"Pkg.add(\"Porta\")","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"or get the latest experimental code.","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"]add https://github.com/iamazadi/Porta.jl.git","category":"page"},{"location":"index.html#Usage","page":"Home","title":"Usage","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"For client-side code read the tests, and for examples on how to build, please check out the models directory. See planethopf.jl as an example.","category":"page"},{"location":"index.html#Status","page":"Home","title":"Status","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"Logic [Doing]\nSet Theory [TODO]\nTopology [TODO]\nTopological Manifolds [TODO]\nDifferentiable Manifolds [TODO]\nBundles [TODO]\nGeometry: Symplectic, Metric [TODO]\nDocumentation [TODO]\nGeometric Unity [TODO]","category":"page"},{"location":"index.html#References","page":"Home","title":"References","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"Physics and Geometry, Edward Witten, (1987)\nThe iconic Wall of Stony Brook University\nThe Road to Reality, Sir Roger Penrose, (2004)\nA Portal Special Presentation- Geometric Unity: A First Look\nPlanet Hopf, Dror Bar-Natan, (2010)\nSPINORS AND SPACE-TIME, Volume 1: Two-spinor calculus and relativistic fields, Roger Penrose, Wolfgang Rindler, (1984)\nA Young Person's Guide to the Hopf Fibration, Zachary Treisman, (2009)\nMathematical Gauge Theory, with Applications to the Standard Model of Particle Physics, Mark J.D. Hamilton, (2018)\nDynamics in the Hopf bundle, the geometric phase and implications for dynamical systems, Rupert Way, (2008)","category":"page"}] +[{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Description = \"How the Hopf fibration works.\"","category":"page"},{"location":"hopffibration.html#The-Hopf-Fibration","page":"Hopf Fibration","title":"The Hopf Fibration","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The Hopf fibration is a fiber bundle with a two-dimensional sphere as the base space and circles as the fiber space. It is the geometrical shape that relates Einstein's spacetime to quantum fields. In this model, we visualize the Hopf fibration by first computing its points via a bundle atlas and then rendering the points in 3D space via stereographic projection. The projection step is necessary because the Hopf fibration is embedded in a four-space. Yet, it has only three degrees of freedom as a three-dimensional shape. The idea that makes this model more special and interesting than a typical visualization is the idea of Planet Hopf, due to Dror Bar-Natan (2010). The basic idea is that since the Hopf map takes the three-dimensional sphere into the two-dimensional sphere, we can pull the skin of the globe back to the three-sphere and visualize it.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Into the bargain, the Earth rotates about its axis every 24 hours. That spinning transformation of the Earth, together with the non-trivial product space of the Hopf bundle, can be encoded naturally into a monolithic visualization. It also makes sense to visualize differential operators in the Minkowski space-time as vectors in a cross-section of the Hopf bundle and then study the properties of spin-transformations. The choice of a gauge transformation (or trivialization) along with Lorentz transformations of Minkowski spacetime should not have any effect on physical laws. It is therefore a great model to understand these transformations and walk the road to reality. The following explains how the source code for generating animations of the Hopf fibration works (alternative views of Planet Hopf). We follow the beginning of chapter 4 of Mark J.D. Hamilton (2018) for a formal definition of the Hopf fibration as a fiber bundle. The book Mathematical Gauge Theory explains the Standard Model to students of both mathematics and physics, covers both the specific gauge theory of the Standard Model and generalizations, and is highly accessible and self-contained. Then, the definitions are going to be used to explain the source code in terms of computational methods and types.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image1) (Image: board2) (Image: image2)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"First, let E and M be smooth manifolds. Then, pi E to M is a surjective and differentiable map between smooth manifolds. Meaning, every element in M has some corresponding element in E via the map pi. Now, let x in M be a point. A fiber of pi over point x is called E_x and defined as a non-empty subset of E as follows: E_x = pi^-1(x) = pi^-1(x) subset E. The singleton of x is taken to the manifold E by the inverse of the map pi. However, to have a set of more than one point let U be a subset of M, U subset M. Then, we have E_U = pi^-1(U) subset E. In this case, E_U is the part of E above the subset U.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image3) (Image: board3) (Image: image4)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, define a global section of the map pi like this: s M to E. Considering the definition of pi E to M, the definition of the global section implies that the composition of pi and s is the identity map pi o s = Id_M over M. A section such as s can be a local one if we take a subset of M in the domain, U subset M. Then, a local section is defined as s U to E. In a similar way the definition of the local section implies that its composition with pi is the idenity map over the subset: pi o s = Id_U. For all points x in subset U, the section s(x) is in the fiber E_x of pi above x, if and only if s is a local section of pi. In this pointwise case, the map pi is restricted to subset U. In other words pi E to U, where U subset M.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image5) (Image: board4) (Image: image6)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In general, for two points x not = y in M that are not equal, the fibers E_x and E_y of pi over x and y may not be embedded submanifolds of E, or even be diffeomorphic. That means, there may not be a differentiable and invertible map that takes fiber E_x into fiber E_y, and the tangent spaces of E_x and E_y over points x and y may not be naturally linear subspaces of the tangent space of E. But, it is different in the special instance where manifold E = M times F is the product of M and the general fiber F and pi as a map is the projection onto the first factor pi M times F to M. If that is the case, then fibers E_x E_y in F of pi over the two distinct points x not = y in M are embedded submanifolds of E and diffeomorphic. To explain it more clearly, given that condition, there exists an invertible and smooth map taking one fiber to the other, and the tangent spaces of the fibers are directly summed with their respective dual subspaces at points in the fibers to span the whole tangent space of manifold E at points of pi over x and y. Therefore, fiber bundles are the generalization of products E = M times F as twisted products.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image7) (Image: board5) (Image: image8)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Before we define the Hopf action, first describe a scalar multiplication rule between vectors and numbers. Let R denote real numbers, Complex complex numbers, and mathbbH quaternionic numbers. On top of that, take a subset of these sets of numbers such that zero is not allowed to be in them, and denote the subsets as R^*, Complex^*, and mathbbH^* respectively. Now, define the linear right action by scalar multiplication for mathbbK = mathbbR mathbbC mathbbH as the following: mathbbK^n+1setminus0 times mathbbK^* to mathbbK^n+1setminus0. For example, 5 in mathbbR^* is a non-zero scalar number, whereas 1 0 0^T in mathbbR^3setminus0 is a non-zero vector quantity. Per our definition, 5 acts on 1 0 0^T on the right and yields 5 0 0 in mathbbR^3setminus0 as another vector. This rule works the same for fields mathbbK even when the vectorial numbers are represented by matrices.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image9) (Image: board6) (Image: image10)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The linear right action by multiplication is called a free action, because for x in mathbbK^n+1setminus0 and y in mathbbk^* the multiplication x times y yields x if and only if y = Id, as the identity element. For example, if we let x = 0 1 0^T y = 1, then the result of the scalar multiplication is 0 1 0^T times 1 = 0 1 0^T.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image11) (Image: board7) (Image: image12)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In addition, we define the unit n-sphere, for the Hopf action works on spheres. So, the unit sphere of dimension n is defined as: S^n(w_1 w_2 w_n+1) in mathbbR^n+1 sum_substack1 leq i leq n+1w_i^2 = 1. As an example, the unit circle S^1 in mathbbC is a one-dimensional sphere with n = 1, and w_1^2 + w_2^2 = 1, where w_1 and w_2 are the horizontal and vertical axes in the complex plane, respectively.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image13) (Image: board8) (Image: image14)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Now, Hopf actions are defined as free actions:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"S^n times S^0 to S^n \nS^2n+1 times S^1 to S^2n+1 \nS^4n+3 times S^3 to S^4n+3 ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"given by (x lambda) mapsto xlambda.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image15) (Image: board9) (Image: image16)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"An example of a Hopf action is the multiplication of the three-sphere S^3 cong SU(2) subset mathbbC^2 on the right by the unit circle S^1 cong U(1) subset mathbbC. Define the Hopf action as the map Phi S^3 times S^1 to S^3 given by (v w lambda) mapsto (v w) sdot lambda = (vlambda wlambda), for all points in the unit 3-sphere (v w) in S^3 and the unit 1-sphere lambda in S^1. What's more, the Hopf action has two properties:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(v w) sdot (lambda sdot mu) = ((v w) sdot lambda) sdot mu\n(v w) sdot 1 = (v w)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"forall (v w) in S^3 lambda mu in S^1.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image17) (Image: board10) (Image: image18)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The next idea is about the orbit of a point in the 3-sphere S^3 under the Hopf action. The orbit map is defined as phi S^1 to S^3 given by lambda mapsto (v_0 w_0) sdot lambda, forall (v_0 w_0) in S^3. The orbit map phi is injective and free, meaning that a point in S^3 can not have many points in S^1 and also there exists an identity element such that the action stabalizes a point in S^3 such as (v_0 w_0). Furthermore, the Hopf action Phi S^1 to Diff(S^3) is a homomorphism. It preserves S^3. The Hopf action being a free action implies that the orbit of every point (v_0 w_0) in S^3 is an embedded circle S^1.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image19) (Image: board11) (Image: image20)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Back to the topic of fiber bundles, we recall that the part of manifold E over subset U equals: E_U = pi^-1(U) subset E, where U subset M. Here, there is an equivalence relation in the fiber E_x of pi over x, since the orbit of a point in fiber E_x by phi collapses onto a single point x in U via the projection map pi S^3 to S^3texttextasciitilde. After the collapse of every fiber in manifold E, the quotient space S^3S^1 is seen to be the projective complex line mathbbCP^1 cong S^2. The projective complex line is the ratio of two complex numbers. To see how the space of S^3 is connected compared to S^1, note that every closed loop in S^3 is shrinkable to a single point in a continuous way, tracing a local section. However, a closed loop in S^1 is not shrinkable to a single point. This fact makes S^3 a simply-connected space and S^1 a not simply-connected space.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image21) (Image: board12) (Image: image22)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We are now almost equipped with the tools to define a fiber bundle in a formal way. Let E F M be manifolds. The projection map pi E to M is a surjective and differentiable map (Every element in M has some element in E). Then, (E pi M F) is called a fiber bundle, (or a locally trivial fibration, or a locally trivial bundle) if for every x in M there exists an open neighborhood U subset M around the point x such that the map pi restricted to E_U can be trivialized as a cross product. Remember that E_U is the part of E of pi over U. In other words, (E pi M F) is called a fiber bundle if there exists a diffeomorphism phi_U E_U to U times F such that pr_1 o phi_U = pi, meaning the projection onto the first factor of the trivialization map phi_U is the same as the map pi. Also, a fiber bundle is denoted by F to E xrightarrowpi M. In this notation, E denotes the total space, M the base manifold, F the general fiber, pi the projection, and (U phi_U) a local trivialization or bundle chart.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image23) (Image: board13) (Image: image24)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Using a local trivialization (U phi_U) E_x = pi^-1(x) we find that the fiber E_x is an embedded submanifold of the total space E for every point x in M. Meaning, the tangent space of fiber E_x is a linear subsapce of the tangent space of E. The direct sum of the tangent subspace of the general fiber and the tangent subspace of the base manifold equals the tangent space of the total space: T_xE = V_xE bigoplus H_xE.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image25) (Image: board14) (Image: image26)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The composition of the local trivialization with the projection onto the second factor gives us yet another useful map between fibers E_x over x and the general fiber F. It is a differentiable and invertible map (diffeomorphism) and equals phi_U = pr_2 o phi_U _E_x E_x to F. Given that the local trivialization phi_U E_U to U times F is a diffeomorphism (invertible and smooth), the projection pr_1 U times F to U onto the first factor of phi_U is a submersion. That is to say the differntial of pr_1 is surjective. D pr_1 T(U times F) to TU takes vectors from the tangent space of U times F into vectors in the tangent space of U, such that every element of TU has some element in T(U times F). As a result, the map pi E to M is also a submersion, which means D pi TE to TM is surjective. Every tangent vector in the codomain TM has some tangent vector in the domain TE.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image27) (Image: board15) (Image: image28)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"So far, we have established that the bundle projection map, taking points from the total space into points in the base space pi E to M, is a submersion. For that reason, the tangent space of the base manifold M is a linear subset of the tangnet space of the total space manifold E. Now, we can use the regular value theorem for shining a light on the submersion of pi. Let a point x in M be a regular value of the smooth map pi E to M, and let the fiber E_x = pi^-1(x) be the preimage of the point x. Then, the map pi^-1 is an embedded submanifold of E of dimension dim E_x = dim E - dim M. Meaning, the tangent space of fiber E_x is a linear subspace of the tangent space of E. We can verify the result of the theorem for the Hopf bundle F to E xrightarrowpi M where dim E = 3 and dim M = 2. The regular value theorem implies that the Hopf fiber is one-dimensional, dim E_x = 3 - 2 = 1, as an embedded submanifold of the total space E. With that formal introduction we are going to sketch a visual 3D model next.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"(Image: image29) (Image: board16) (Image: image30) (Image: image31)","category":"page"},{"location":"hopffibration.html#Import-the-Required-Packages","page":"Hopf Fibration","title":"Import the Required Packages","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Begin by importing a few software packages for doing algebraic operations, working with files and graphics processing units. Besides Porta, we need to use three packages: FileIO, GLMakie and LinearAlgebra. First, FileIO is the main package for IO and loading all different kind of files, including images and Comma-Separated Value (CSV) files. Second, interactive data visualizations and plotting in Julia are done with GLMakie. Finally, LinearAlgebra, as a module of the Julia programming language, provides array arithmetic, matrix factorizations and other linear algebra related functionality. However, through years of working with geometrical structures and shapes we have encapsulated certain mathematical computations and transformations into custom types and interfaces, which make up most of the functionalities of project Porta. In addition, we wrapped complicated computer graphics workflows inside custom types in order to increase the interoperability of our types with those of external packages such as GLMakie.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" import FileIO\n import GLMakie\n import LinearAlgebra\n using Porta","category":"page"},{"location":"hopffibration.html#Set-Hyperparameters","page":"Hopf Fibration","title":"Set Hyperparameters","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"There are essential hyperparameters that determine the complexity of graphics rendering as well as the position and orientation of a camera, through which we render a scene. Since the output of the model is an animation video, we need to set the figure size to 1920 by 1080 to have a full high definition window, in which the scene is located. Most of the shapes and objects that we put inside of the scene are two-dimensional surfaces. Therefore, the segmentation of most shapes requires two integer values for determining how much compute power and resolution we are willing to spend on the animation. Furthermore, the shape of a circle is the most common in our scenes because of the magic of complex numbers. It is known that using 30 segments results in smooth low-polygon circles. So for a two-dimensional sphere a 30 by 30 segmented two-surface should look good. Set the segments equal to 30, and less curvy shapes will look even better in consequence. But, an animation extends through time frame by frame and so we need to set the total number of frames. In this way, specifying the number of frames determines the length of the video. For example, 1440 frames make a one-minute video at 24 frames per second.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" figuresize = (1920, 1080)\n segments = 30\n frames_number = 1440","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A model means a complicated geometrical shape contained inside a graphical scene. Every model has a name to use as the file name of the output video. Here, we choose the name planethopf as we construct an alternative view of the Planet Hopf by Dror Bar-Natan (2010). Heinz Hopf in 1931 discovered a way to join circles over the skin of the globe. The discovery defines a fiber bundle where the base space is the spherical Earth and the fibers are circles. But, the circles are all mutually parallel and linked. Moreover, the Earth goes through a full rotation about the axis that connect the poles every 24 hours. So it is not surprising that the picture of a non-trivial bundle and the spinning of the base space coordinates (longitudes) makes for a ridiculous geometric shape. But, the surprising fact is that all of it is visualizable as a 3D object. Then, we use a dictionary that maps indices to names in order to keep track of boundary data on the globe and the name of each boundary as a sovereign country.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" modelname = \"planethopf\"\n indices = Dict()","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The Hopf fibration, as a fiber bundle, has an inner product space. The inner product space is symmetric, linear and positive semidefinite. The last property means that the product of a point in the bundle with itself is always non-negative, and it is zero if and only if the point is the zero vector. The abstract inner product space allows us to talk about the length of vectors, the distance between two points and the idea of orthogonality between two vectors. A pair of vectors are orthogonal when they make a right angle with each other and as a consequence their product is equal to zero. For all u v v_1 v_2 in V and alpha beta in R the following are the properties of the abstract inner product space:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Symmetric: u v = v u\nLinear: u alpha v_1 + beta v_2 = alpha u v_1 + beta u v_2\nPositive semidefinite: u u geq 0 for all u in V with u u = 0 if and only if u = 0","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Now, in order to skin the horizontal cross-sections of the bundle for visualization we need to start with a base point, which is denoted by x. At the tangent space of the base point q, the inner product space (characterized by a connection one-form) splits the tangent space of the bundle E at x into two linear subspaces: horizontal and vertical.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"T_q E = V_q E bigoplus H_q E","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In terms of the connection, the two subspcaes are orthogonal. A chart is a four-tuple of real numbers to be used as a pair of closed intervals in the horizontal subspace. Then, using the exponential map one can travel in both horizontal and vertical directions and cover the whole bundle within the lengths of the chart intervals. Within the boundary of the chart and with an additional vertical coordinate (a gauge) we can define a tubular neighborhood of the base point q. The first two elements of the four-tuple chart give the interval along the first basis vector and the last two elements give the interval along the second basis vector. As for the third basis vector of the tangent space (the vertical subspace) we use a beginning and an ending gauge.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"For the purpose of the construction of the Hopf fibration we define the bundle atlas of a general fiber bundle F to E xrightarrowpi M as an open covering U_i_i in I of the base manifold M together with bundle charts phi_i E_U_i to U_i times F. Putting the open covering with bundle charts a bundle atlas is denoted by U_i phi_i_i in I. The index i suggests that a bundle atlas should have more than one bundle chart whenever it is a non-trivial bundle (a twisted product rather than a Cartesian product). In order to cover the Hopf bundle we use the exponential matrix function supplied with linear combinations of elements from the Lie algebra so(4), which produces elements in the Lie group SO(4) that push a base point around the 3-sphere. As a side note, a Lie algebra is a vector space V that is equipped with the Lie bracket map sdot sdot V times V to V, with sdot sdot having three properties: bilinear, antisymmetric and satisfies the Jacobi identity. We choose a base point in the 3-sphere q in S^3 and then use Lie algebra elements before exponentiation in order to rotate the 3-sphere to cover every other point in the total space S^3 over the chart.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" q = Quaternion(ℝ⁴(0.0, 0.0, 1.0, 0.0))\n chart = (-π / 4, π / 4, -π / 4, π / 4)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, we define five scalars in the Lie algebra of so(2), identified with imathbbR, in order to provide different gauge transformations for pullbacks by the Hopf fibration (whirls and base maps). The exponential function takes the gauge values to the unit circle S^1 = U(1) cong SO(2) given by exp(im * gauge). For creating a clearer view we are going to slice up the Hopf fibers (orbits) and set different values for their respective alpha channels. The names gauge1, gauge2, gauge3, gauge4 and gauge5 are used to provide the Hopf actions when we construct and update the shapes. 0.0 means the trivial action whereas 2π means the full orbit around a Hopf fiber. Looking at the values of these names we can see that a Hopf fiber will be cut into four quarters. We can make some quarters opaque and others see-through for better visibility.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" gauge1 = 0.0\n gauge2 = π / 2\n gauge3 = float(π)\n gauge4 = 3π / 2\n gauge5 = 2π","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The fundamental physics is based on the gauge symmetry of the product SU(3) times SU(2) times U(1) and the symmetry of spacetime as a Riemannian manifold M that is equipped with a metric. Therefore, physical laws in nature must be the same under two sets of choices: the choice of gauge transformations and the choice of an inertial reference frame in spacetime. In this model, we understand the choice of the guage symmetry by studying the Hopf action and the choice of an inertial frame in Minkowski space-time by a change-of-basis transformation on the Hopf bundle. The change-of-basis transformation is denoted by matrix M and is applied to the total space of the Hopf bundle via a matrix-vector product. Here, we initialize the matrix M with the idenity.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" M = I(4)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In order to get the essence of these different choices and integrate them into a visual model we first note that Lorentz transformations of null vectors in the tangent space of spacetime is the same as transforming any other timelike (non-null) vectors. Second, The Hopf bundle of the 3-sphere has a representation in the Lie group S^3 = SU(2) and the Hopf action is represented by actions of S^1 = U(1) as a linear scalar multiplication on the right. But, null vectors have length zero in terms of the Lorentzian metric, whereas the Hopf bundle is made of vectors of unit length in terms of the Euclidean metric. Fortunately, these vectors coincide as unit quaternions and so their transformations can be unified into a single visual model. If we coordinatize a null vector in spacetime as u = 𝕍(T, X, Y, Z) then the corresponding quaternion q = Quaternion(T, X, Y, Z) takes the same coordinates. We assert that u is null and q is of unit norm, with an approximate equality check. The precision of the assertion is given by the name tolerance, which equals 1e-3.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" T, X, Y, Z = vec(normalize(ℝ⁴(1.0, 0.0, 1.0, 0.0)))\n u = 𝕍(T, X, Y, Z)\n q = Quaternion(T, X, Y, Z)\n tolerance = 1e-3\n @assert(isnull(u, atol = tolerance), \"u in not a null vector, $u.\")\n @assert(isapprox(norm(q), 1, atol = tolerance), \"q in not a unit quaternion, $(norm(q)).\")","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The camera is a viewport trough which we see the scene. It is a three-dimensional camera and much like a drone it has six features to help position and orient itself in the scene. Accordingly, a three-vector in the Euclidean 3-space E^3 determins its position in the scene, another 3-vector specifies the point at which it looks, and a third vector controls the up direction of the camera. The third 3-vector is needed because the camera can rotate through 360 degrees about the axis that connects its own position to the position of the subject. Using these three 3-vectors we control how far away we are from the subject, and how upright the subject is. ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" eyeposition = normalize(ℝ³(1.0, 1.0, 1.0)) * π * 0.8\n lookat = ℝ³(0.0, 0.0, 0.0)\n up = normalize(ℝ³(1.0, 0.0, 0.0))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Each of the eyeposition, lookat and up vectors are in the three-real-dimensional vector space ℝ³. The structure of the abstract vector space of ℝ³ includes: associativity of addition, commutativity of addition, the zero vector, the inverse element, distributivity Ι, distributivity ΙΙ, associativity of scalar multiplication, and the unit scalar 1. Also, the product space associated with ℝ³ is symmetric, linear and positive semidefinite (see real3_tests.jl). The same goes for the structure of 4-vectors in ℝ⁴ as we are going to encounter in this model. An abstract vector space (V mathbbK + ) consists of four things:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A set of vector-like objects V = u v \nA field mathbbK of scalar numbers, complex numbers, quaternions, or octonions (any one of the division algebras)\nAn addition operation + for elements of V that dictates how to add vectors: u + v\nA scalar multiplication operator for scaling a vector by an element of the field","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"An abstract vector space satisfies eight axioms. For all vectors u v w in V and for all scalars alpha beta in mathbbK the following properties are true:","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Associativity of addition: u + (v + w) = (u + v) + w\nCommutativity of addition: u + v = v + u\nThere exists a zero vector 0 in V such that u + 0 = 0 + u = u\nFor every u there exists an inverse element -u such that u + (-u) = u - u = 0\nDistributivity I: alpha (u + v) = alpha u + alpha v\nDistributivity II: (alpha + beta) u = alpha u + beta u\nAssociativity of scalar multiplication: alpha (beta u) = (alpha beta) u\nThere exists a unit scalar 1 such that 1u = u","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Interestingly, if the field mathbbK is an Octonian number then the axiom of the commutativity of addition becomes false. The plan is to first load a geographic data set, then construct a few shapes, and animate a four-stage transformation of the shapes. Model versioning can be applied here using different stages. The transformations are subgroups of the Lorentz transformation in the Minkowski vector space 𝕍, which is a tetrad and origin point away from the Minkowski space-time 𝕄. Both 𝕍 and 𝕄 inherit the properties of the abstract vector space. See minkowskivectorspace_tests.jl and minkowskispacetime_tests.jl for use cases.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" totalstages = 4","category":"page"},{"location":"hopffibration.html#Load-the-Natural-Earth-Data","page":"Hopf Fibration","title":"Load the Natural Earth Data","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, we need to load two image files: an image to be used as a color reference, and another one to be used as surface texture for sections of the Hopf bundle. This is the first example of using FileIO to load image files from hard drive memory. Both images are made with a software called QGIS, which is a geographic information system software that is free and open-source. But, the data comes from Natural Earth Data. Natural Earth is a public domain map dataset available at 1:10m, 1:50m, and 1:110 million scales. Featuring tightly integrated vector and raster data, with Natural Earth you can make a variety of visually pleasing, well-crafted maps with cartography or GIS software. We downloaded the Admin 0 - Countries data file from the 1:10m Cultural Vectors link of the Downloads page. It is a large-scale map that contains geometry nodes and attributes.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" reference = FileIO.load(\"data/basemap_color.png\")\n mask = FileIO.load(\"data/basemap_mask.png\")","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"As for the image files, we paint the boundaries using the gemometry nodes, and add a grid to be able to visualize distortions of the Euclidean metric of the underlying surface. Therefore, the reference is the clean image from which we pick colors, whereas the mask has a grid and transparency for visualization purposes.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" attributespath = \"data/naturalearth/geometry-attributes.csv\"\n nodespath = \"data/naturalearth/geometry-nodes.csv\"\n countries = loadcountries(attributespath, nodespath)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The geometry nodes of the data set consist of latitudes and longitudes of boundaries. But, geometry attributes feature various geographical, cultural, economical and geopolitical values. Of these features we only need the names and geographic coordinates. To not limit the use cases of this model, the generic function loadcountries loads all of the data features by supplying it with the file paths of attributes and nodes. Data versioning can be applied here using different file versions. The attributes and nodes files are comma-separated values.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"At a high level of description, the process of loading boundary data is as follows: First, we use FileIO to open the attributes file. Second, we put the data in a DataFrames object to have in-memory tabular data. Third, sort the data according to shape identification. Fourth, open the nodes file in a DataFrame. Fifth, group the attributes by the name of each sovereign country. Sixth, determine the number of attribute groups by calling the generic function length. Seventh, define a constant ϵ = 5e-3 to limit the distance between nodes so that the computational complexity becomes more reasonable. Eighth, define a dictionary that has the keys: shapeid, name, gdpmd, gdpyear, economy, partid, and nodes. Finally, for each group of the attributes we extract the data corresponding to the dictionary keys and push them into array values.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Part of the difficulty with the data loading process is that each sovereign country may have more than one connected component (closed boundary). That is why we store part identifications as one of the dictionary keys. In this process, the part with the greatest number of nodes is chosen as the main part and is pushed into the corresponding array value. All of the array values are ordered and have the same length so that indexing over the values of more than one key becomes easier. Once the part ID of each country name is determined, we make a subset of the data frame related to the part ID and then extract the geographic coordinates in terms of latitudes and longitudes. In fact, we make a histogram of each unique part ID and count the number of coordinates. The part ID with the greatest number of coordinates is selected for creating the subset of the data frame. Next, the coordinates are transformed into the Cartesian coordinate system from the Geographic one.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Finally, we decimate a curve containing a sequence of coordinates by removing points from the curve that are farther from each other than the given threshold ϵ. It is a step to make sure that the boundary data has superb quality while managing the size of data for computation complexity. The generic function decimate implements the Ramer–Douglas–Peucker algorithm. It is an iterative end-point fit algorithm suggested by Dror Bar-Natan (2010) for this model. Since a boundary is modelled as a curve of line segments, we set a segmentation limit. But, the decimation process finds a curve that is similar in shape, yet has fewer number of points with the given threshold ϵ. In short, decimate recursively simplifies the segmented curve of a closed boundary if the maximum distance between a pair of consecutive points is greater than ϵ. The distance between two abstract vectors is given by d(u v) equiv u - v = sqrt(u - v) (u - v).","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" boundary_names = [\"United States of America\", \"Antarctica\", \"Australia\", \"Iran\", \"Canada\", \"Turkey\", \"New Zealand\", \"Mexico\", \"Pakistan\", \"Russia\"]\n boundary_nodes = Vector{Vector{ℝ³}}()\n for i in eachindex(countries[\"name\"])\n for name in boundary_names\n if countries[\"name\"][i] == name\n push!(boundary_nodes, countries[\"nodes\"][i])\n println(name)\n indices[name] = length(boundary_nodes)\n end\n end\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"As the boundary data is massive in number (248 countries) we need to select a subset for visualization. 10 countries selected from a linear space of alphabetically sroted names should be representative of the whole Earth. Then again, using only three distinct points in the 2-sphere one can infer the transformations from the sphere into itself. Also, Antarctica should be added due to its special coordinates at the south pole, to give the user a better sense of how bundle sections are expanded and distorted. As soon as we have the names of the selection, we can proceed with populating the dictionary of indices that relates the name of each country with the corresponding index in boundary data. Using the dictionary we can read the attributes of countries by giving just the name as argument.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" points = Vector{Quaternion}[]\n for i in eachindex(boundary_nodes)\n _points = Quaternion[]\n for node in boundary_nodes[i]\n r, θ, ϕ = convert_to_geographic(node)\n push!(_points, q * Quaternion(exp(ϕ / 4 * K(1) + θ / 2 * K(2))))\n end\n push!(points, _points)\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We instantiate a vector of a vector of type Quaternion to store boundary data. The outermost vector contains elements of different countries. But, the innermost vector contains the pullback of the geographic nodes by the Hopf map in the 3-sphere. After conversion to the Geographic coordinate system from the Cartesian coordinates, the points are pulled back by pi using the statement q * Quaternion(exp(ϕ / 4 * K(1) + θ / 2 * K(2))). It is a right multiplication of the base point q by the exponential function, supplied with the geographic coordinates θ and ϕ. Now that we have the points we can make a 3D scene.","category":"page"},{"location":"hopffibration.html#Make-a-Computer-Graphical-Scene","page":"Hopf Fibration","title":"Make a Computer Graphical Scene","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Scenes are fundamental building blocks of GLMakie figures. In this model, the layout of the Figure (graphical window) is a single Scene, because we have been able to directly plot all of the information about the bundle geometry and topology inside the same scene. The figure is supplied with the hyperparameter figuresize that we defined earlier. Then, we set a black theme to have black background around the window at the margins. Next, we instantiate a gray point light and a lighter gray ambient light. The lights together with the figure are then passed to LScene to construct our scene. We pass the symbol :white as the argument to the background keyword as it makes for the most visible scene.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" makefigure() = GLMakie.Figure(size = figuresize)\n fig = GLMakie.with_theme(makefigure, GLMakie.theme_black())\n pl = GLMakie.PointLight(GLMakie.Point3f(0), GLMakie.RGBf(0.0862, 0.0862, 0.0862))\n al = GLMakie.AmbientLight(GLMakie.RGBf(0.9, 0.9, 0.9))\n lscene = GLMakie.LScene(fig[1, 1], show_axis=false, scenekw = (lights = [pl, al], clear=true, backgroundcolor = :white))","category":"page"},{"location":"hopffibration.html#Construct-Base-Maps","page":"Hopf Fibration","title":"Construct Base Maps","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The base map is the pullback of the skin of the globe U subset S^2 by the Hopf map pi S^3 to S^2, representing a local horizontal cross-section of the bundle. The pushforward of horizontal vectors by the Hopf map leaves them unchanged. However, vectors in the vertical subsapce of the tangent space of the Hopf bundle are in the kernel of the Hopf map (they are sent to zero).","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We use a 64-bit floating point number to parameterize an element of the Lie algebra so(2), before exponentiating it into an element of the Lie group SO(2) to be used for the orbit map phi S^1 to S^3, because a local horizontal cross-section uses the same scalar number for the entirety of subset U subset S^2. The subset U is bounded with a two-dimensional chart. A chart can be thought of as a rectangle whose sides are at most π in length. But, the length of a great circle of the three-dimensional sphere is 2π and the maximum length of chart sides is limited, unless we want to cover S^3 twice. To keep things simple, we use one bundle chart and cover a subset U of side length π. The Hopf bundle does not admit a global section. After exponentiating the base point q in horizontal directions for a magnitude beyond π, the orientation of the surface reverses and a sharp twist of the surface happens.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The Hopf bundle is embedded in ℝ⁴, the real-four-dimensional space. The coordinates are defined as unit quaternions where the basis vectors are represented by the symmetry group of the rotations of an orthogonal tetrad, namely SO(4). vectors u and v are orthogonal if and only if their inner product equals zero u v = 0. When we talk about Hopf actions and bundle charts, we talk about values that are used to linearly combine elements of the Lie algebra of so(4), vectors in the tangent space of the bundle at point x. Then, we use the matrix exponential map for computing Lie group values in SO(4). Given a fixed gauge, a point in the Lie group stemming from base point x is reconstructed from a Lie algebra element by executing the statement x * Quaternion(exp(θ * K(1) + -ϕ * K(2)) * exp(gauge * K(3))), where scalars θ and ϕ denote the latitude and longitude components in the bundle chart, respectively. K(1) and K(2) denote 4x4 matrices with real elements as basis vectors of the Lie algebra so(4). The tangent space of the bundle at point x spans horizontally with the exponential map of a linear combination of basis vectors K(1) and K(2), whereas it spans vertically in the K(3) direction. This way we get a strictly horizontal section of the bundle in terms of elements of the Lie group SO(4), given a gauge. The elements of SO(4) go on to push the base point x around and end up as observables to be rendered graphically.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" lspaceθ = range(chart[1], stop = chart[2], length = segments)\n lspaceϕ = range(chart[3], stop = chart[4], length = segments)\n [project(normalize(M * (x * Quaternion(exp(θ * K(1) + -ϕ * K(2)) * exp(gauge * K(3)))))) for ϕ in lspaceϕ, θ in lspaceθ]","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Using the eigendecomposition method LinearAlgebra.eigen, we can compute the matrix M to change the basis of the bundle while keeping the coordinates invariant. So the change-of-basis is the final step of the construction of the observables after using the geographic coordinates and the gauge. Observables.jl allows us to define the points that are to be rendered in the scene, in a way that they can listen to changes dynamically. Later, when we apply transformations to the bundle, including the change-of-basis, the idea is to only change the top-level observables and avoid reconstructing the scene entirely. The change of basis is a bilinear transformation of the tetrad (of Minkowski space-time 𝕄) in ℝ⁴ as a matrix-vector product (M * x for example). Here we denote the transformation as matrix M, which takes a Quaternion number as input and spits out a new number of the same type. The input and output bases must be orthonormal as the numbers must remain unit quaternions after the transformation. Constructing a base map requires a few arguments: the scene object, the base point q, the gauge, the change-of-basis transformation M, the chart, the number of segments of the lattice of observables, the tuxture of the surface and the optional transparency setting. Construct four base maps in order to visualize a more complete picture of the Hopf fibration using four different sections. But, the sections are going to be distinguished from one another and updated with gauge transformations later when we animate them.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" basemap1 = Basemap(lscene, q, gauge1, M, chart, segments, mask, transparency = true)\n basemap2 = Basemap(lscene, q, gauge2, M, chart, segments, mask, transparency = true)\n basemap3 = Basemap(lscene, q, gauge3, M, chart, segments, mask, transparency = true)\n basemap4 = Basemap(lscene, q, gauge4, M, chart, segments, mask, transparency = true)","category":"page"},{"location":"hopffibration.html#Construct-Whirls","page":"Hopf Fibration","title":"Construct Whirls","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A Whirl is the shape of a closed boundary in the map of the Earth that is pulled back by the Hopf map pi S^3 to S^2. As a reminder, boundaries on the map of the Earth are specified by two real values: latitude θ and longitude ϕ. The boundary of each country in boundary_names is lifted up from the base manifold using the following statement: q * Quaternion(exp(ϕ / 4 * K(1) + θ / 2 * K(2))). The pullback operation is realized by pushing the base point q in a horizontal direction given by coordinates on the surface of the Earth. Then, a gauge transformation is applied by executing the statement x * Quaternion(exp(K(3) * gauge)), with the given scalar gauge in the direction K(3) of the tangent space at point x of the bundle. By varying gauge in a linear space of floating point values, a Whirl (a pullback by the Hopf map) takes a three-dimensional volume. In the special case where gauge is a range of values, starting at zero and stopping at 2π, the Whirl makes a Hopf band. The degree of the twist in the band is directly proportional to the value of gauge. Multiplying x on the right by the exponentiation of K(3) * gauge pushes x in the vertical subspace of the bundle and makes an orbit. Therefore, the orbit map phi S^1 to S^3 is given by x[i] * Quaternion(exp(K(3) * gauge).","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" lspacegauge = range(gauge1, stop = gauge2, length = segments)\n [project(normalize(M * (x[i] * Quaternion(exp(K(3) * gauge))))) for i in 1:length(x), gauge in lspacegauge]","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"There are four sets of whirls: some whirls are more solid and some whirls are more transparent. This separation is done to highlight the antipodal points of the three-dimensional sphere S^3, given by x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1, where x_1 x_2 x_3 x_4^T in R^4. It also helps to visualize the direction of the null plane under transformations of the bundle. Since every pair of points that are infinitestimally close to each other in a horizontal cross-section, defines a differential operator. And Hopf actions, transformations from the bundle into itself change the direction of the operator as it twists. The operator is also called a spin-vector in Minkowski vector space 𝕍. Therefore it can be visualized directly how the operator changes sign by comparing a pullback into S^3 at antipodal points of an orbit.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" whirls1 = []\n whirls2 = []\n whirls3 = []\n whirls4 = []\n for i in eachindex(boundary_nodes)\n color1 = getcolor(boundary_nodes[i], reference, 0.1)\n color2 = getcolor(boundary_nodes[i], reference, 0.2)\n color3 = getcolor(boundary_nodes[i], reference, 0.3)\n color4 = getcolor(boundary_nodes[i], reference, 0.4)\n whirl1 = Whirl(lscene, points[i], gauge1, gauge2, M, segments, color1, transparency = true)\n whirl2 = Whirl(lscene, points[i], gauge2, gauge3, M, segments, color2, transparency = true)\n whirl3 = Whirl(lscene, points[i], gauge3, gauge4, M, segments, color3, transparency = true)\n whirl4 = Whirl(lscene, points[i], gauge4, gauge5, M, segments, color4, transparency = true)\n push!(whirls1, whirl1)\n push!(whirls2, whirl2)\n push!(whirls3, whirl3)\n push!(whirls4, whirl4)\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The color of a Whirl should match the color of the inside of its own boundary at every horizontal section, also known as a base map. The generic function getcolor finds the correct color to set for the Whirl. It takes as input a closed boundary (a vector of Cartesian points), a color reference image and an alpha channel value to produce an RGBA 4-color. getcolor finds a color according to the following steps: First, it determines the number of points in the given boundary. Second, gets the size of the reference color image as height and width in pixels. Third, converts all of the boundary points to Geographic coordinates. Fourth, finds the minimum and maximum values of the latitudes and longitudes of the boundary. Fifth, creates a two-dimensional linear space (a flat grid or lattice) that ranges within the upper and lower bounds of the latitudes and longitudes. Sixth, finds the Cartesian two-dimensional coordinates of the points in the image space by normalizing the geographic coordinates and multiplying them by the image size. Seventh, picks the color of each grid point with the Cartesian two-dimensional coordinates in the image space as the index. Eighth, Makes a histogram of the colors by counting the number of each color. Finally, sorts the histogram and picks the color with the greatest number of occurance. (See earth.jl from the src directory for implementation.)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"However, step seven makes sure that the coordinates in the linear 2-space are inside the closed boundary, otherwise it skips the index and continues with the next index in the grid. In this way we don't pick colors from the boundaries of neighboring countries over the globe. The generic function isinside is used by getcolor to determine whether the given point is inside the given boundary or not. But first, the boundary needs to become a polygon in the Euclidean 2-space of coordinates in terms of latitude and longitude. This is the same as geographic coordinates with the radius of Earth set equal to 1 identically, hence the spherical Earth model of the ancient Greeks. After we make a polygon out of the boundary, the generic function rayintersectseg determines whther a ray cast from a point of the linear grid intersects an edge with the given point p and edge. Here, p is a two-dimensional point and edge is a tuple of such points, representing a line segment. Eventhough this algorithm should work in theory, some boundaries are too small to yield a definite color via getcolor and the color inference algorithm returns a false negative in those cases. So the default color may be white for a limited number of cases out of 248 countries. Once we have the color of the whirls, we can proceed to construct the whirls by supplying the generic function Whirl with the following arguments: the scene object, the boundary points lifetd via an arbitrary section, the first fiber action value (gauge), the second action value, the change-of-basis function M, the number of surface segments, the color and the optional transparency setting.","category":"page"},{"location":"hopffibration.html#Compute-a-Four-Screw","page":"Hopf Fibration","title":"Compute a Four-Screw","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We are going to execute a motion around a closed loop in the Lie group SL(2 mathbbC), and then multiply every point in the Hopf bundle by an element of the loop. A four-screw is a subset of the Lie group SL(n mathbbK) = A in Mat(n times n mathbbK) det(A) = 1, square matrices of Complex numbers whose volume form (determinant) equals 1. Here, the number n = 2 and the field mathbbK = mathbbC. A four-screw is a kind of restricted Lorentz transformation where a z-boost and a proper rotation of the celestial sphere are applied. The transformation lives in a four-complex dimensional space and it has six degrees of freedom (the same number of dimensions as SO(4)). By parameterizing a four-screw one can control how much boost and rotation a transformation shuld have. Here, w as a positive scalar controls the amount of boost, whereas angle ψ controls the rotation component of the transform. But, the parameterization accepts rapidity as input for the boost. So we take the natural logarithm of w (log(w) = phi) in order to supply the transformer with the required rapidity argument. First, we set w equal to one in order to preserve the scale of the Argand plane and animate the angle ψ through zero to 2π for rotation. The name progress denotes a scalar from zero to one for instantiating a different transformation at each frame of the animation.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" if status == 1 # roation\n w = 1.0\n ϕ = log(w) # rapidity\n ψ = progress * 2π\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"In the second case, we fix the rotation angle ψ by setting it to zero, and this time animate the rapidity by changing the value of ϕ at each time step.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" if status == 2 # boost\n w = max(1e-4, abs(cos(progress * 2π)))\n ϕ = log(w) # rapidity\n ψ = 0.0\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Third, in order to get a complete picture of a four-screw we animate both rapidity ϕ and rotation ψ, at the same time.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" if status == 3 # four-screw\n w = max(1e-4, abs(cos(progress * 2π)))\n ϕ = log(w) # rapidity\n ψ = progress * 2π\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A four-real-dimensional vector in the Minkowski vector space 𝕍 is null if and only if its Lorentz norm is equal to zero. The length or norm of an abstract vector u in V is equivalent to the square root of the inner product of the vector with itself: u u equiv sqrtu u in R. The inner product of vectors u and v in an abstract vector space is given by u^T * g_munu * v, where g_munu denotes the metric 2-tensor. However, as an instantiation in Minkowski vector space 𝕍 with signature (+, -, -, -), the matrix g_munu is a diagonal of the form: g_munu = beginbmatrix 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 -1 endbmatrix.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Furthermore, a vector in 𝕍 is in the tangent space at some point in Einstein's spacetime, where the metric g_munu will not be diagonal in general. Since a Lorentz transformation of null vectors has the same effect on vectors that are not null, it makes the visualization easier to study transformations on null vectors only. On the other hand, in the Euclidean 4-space E^4 the metric g_munu is replaced by identity matrix of dimension four. The null vectors that we use here in the Minkowski vector space have length zero in terms of the Lorentz norm, but have Euclidean norm equal to one, and so they can be regarded as elements of unit Quaternion. Therefore, what we are animating here is the transformation of unit quaternions that represent null vectors. ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The change-of-basis transformations that we have used to instantiate Whirl and Basemap types above, can accomodate the effects of a Lorentz transformation. Then, by setting ψ and ϕ we can define a generic function transform to take Quaternion numbers as input and to give us the transformed number as output.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" transform(x::Quaternion) = begin\n T, X, Y, Z = vec(x)\n X̃ = X * cos(ψ) - Y * sin(ψ)\n Ỹ = X * sin(ψ) + Y * cos(ψ)\n Z̃ = Z * cosh(ϕ) + T * sinh(ϕ)\n T̃ = Z * sinh(ϕ) + T * cosh(ϕ)\n Quaternion(T̃, X̃, Ỹ, Z̃)\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Every transformation in an abstract vector space such as the Minkowski vector space 𝕍 has a matrix representation. For constructing the matrix of the transform we just need to compute it four times with basis vectors. The transformation of the basis vectors of unit quaternions by transform are denoted by r₁, r₂, r₃ and r₄. The matrix _M is a four by four real matrix whose rows are r₁ through r₄. _M is the matrix representation of the transformation that is induced by transform.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" r₁ = transform(Quaternion(1.0, 0.0, 0.0, 0.0))\n r₂ = transform(Quaternion(0.0, 1.0, 0.0, 0.0))\n r₃ = transform(Quaternion(0.0, 0.0, 1.0, 0.0))\n r₄ = transform(Quaternion(0.0, 0.0, 0.0, 1.0))\n _M = reshape([vec(r₁); vec(r₂); vec(r₃); vec(r₄)], (4, 4))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"But, _M doesn't necessarily take unit quaternions to unit quaternions. By decomposing _M into eigenvalues and eigenvectors we can manipulate the transformation so that it takes unit quaternions to unit quaternions without modifying its effect on the geometrical structure of Argand plane. Despite the fact that _M is a matrix of real numbers, it has complex eigenvalues, as it involves a rotation. By constructing a four-complex-dimensional vector off of the eigenvalues we can normalize _M by normalizing the vector of eigenvalues, before reconstructing a unimodular, unitary transformation (a normal matrix). The reconstructed matrix is called M.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" decomposition = LinearAlgebra.eigen(_M)\n λ = LinearAlgebra.normalize(decomposition.values) # normalize eigenvalues for a unimodular transformation\n Λ = [λ[1] 0.0 0.0 0.0; 0.0 λ[2] 0.0 0.0; 0.0 0.0 λ[3] 0.0; 0.0 0.0 0.0 λ[4]]\n M = real.(decomposition.vectors * Λ * LinearAlgebra.inv(decomposition.vectors))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"We can assert that the transformation that is induced by M takes null vectors to null vectors in Minkowski vector space 𝕍. If that is the case, then the reconstructed transformation M is a faithful representation and it only scales the extent of null vectors rather than null directions, compared to _M. A representation f is called a faithful representation when for different numbers g and q, f(g) and f(q) are equal if and only if g = q.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A spin-vector is based on the space of future or past null directions in Minkowski space-time. The field ζ of a SpinVector represents points in Argand plane. Therefore, if v is obtained with the transformation of u by M, then the respective spin-vectors s and s′ should tell us how M changes Argand plane. To be precise, three different points in Argand plane, namely u₁, u₂, u₃, are needed to characterize the transformation. We assert that the transformation by M induced on Argand plane is correct, because it extends the Argand plane ζ = w * exp(im * ψ) * s.ζ by magnitude w and rotates it through angle ψ. So, we established the fact that normalizing the vector of eigenvalues of the transformation _M and reconstructing it to get M leaves the effect on Argand plane invariant.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" u₁ = 𝕍(1.0, 1.0, 0.0, 0.0)\n u₂ = 𝕍(1.0, 0.0, 1.0, 0.0)\n u₃ = 𝕍(1.0, 0.0, 0.0, 1.0)\n for u in [u₁, u₂, u₃]\n v = 𝕍(vec(M * Quaternion(u.a)))\n @assert(isnull(v, atol = tolerance), \"v ∈ 𝕍 in not null, $v.\")\n s = SpinVector(u)\n s′ = SpinVector(v)\n if s.ζ == Inf # A Float64 number (the point at infinity)\n ζ = s.ζ\n else # A Complex number\n ζ = w * exp(im * ψ) * s.ζ\n end\n ζ′ = s′.ζ\n if ζ′ == Inf\n ζ = real(ζ)\n end\n @assert(isapprox(ζ, ζ′, atol = tolerance), \"The transformation induced on Argand plane is not correct, $ζ != $ζ′.\")\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"A distinction between coordinates in Argand plane becomes relevant when we want to assert the properties of M on a test variable ζ, without applying M on a control variable ζ′. In the special case where the null direction ζ is the point at infinity, the north pole, we expect for the transformation induced by M to be inconsequential. Because ζ is a union of complex numbers and the singleton of infinity (of type Union{Complex, ComplexF64, Float64}). For an inhomogeneous coordinate system we treat the point at infinity in a different way. For example, for all values of w, if ζ equals infinity then the rotation component of a four-screw should not have any effect on the north pole. But, multiplying positive infinity by a complex number of negative magnitude makes ζ equal to negative infinity, which is not in Argand plane. In that case, we first check the edge case to leave ζ unchanged whenever its value is infinity, ζ = s.ζ. No amount of z-boost and rotation about the z-axis should transform the north pole. Else, ζ transforms as expected: ζ = w * exp(im * ψ) * s.ζ.","category":"page"},{"location":"hopffibration.html#Compute-a-Null-Rotation","page":"Hopf Fibration","title":"Compute a Null Rotation","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"To understand a null rotation, imagine that you are an astronaut in empty space, far away from any celestial object. Looking at the space around you from every direction, you can see your surrounding environment through a spherical viewport. This view is called the celestial sphere of past null directions, as the light from the stars in the past reach your eyes. A null rotation translates Argand plane such that just one null direction is invariant, the point at infinity (the north pole of the celestial sphere). We control the animation of a null rotation by defining a real number a.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" a = sin(progress * 2π)","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Whenever T is positive, we talk about the sphere of future-pointing null directions. At this stage of the animation, the transformation transform defines a null rotation such that the invariant null vector is the direction t + z, the north pole of the sphere of future-pointing null directions, where ζ equals infinity. ","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" transform(x::Quaternion) = begin\n T, X, Y, Z = vec(x)\n X̃ = X \n Ỹ = Y + a * (T - Z)\n Z̃ = Z + a * Y + 0.5 * a^2 * (T - Z)\n T̃ = T + a * Y + 0.5 * a^2 * (T - Z)\n Quaternion(T̃, X̃, Ỹ, Z̃)\n end\n\n r₁ = transform(Quaternion(1.0, 0.0, 0.0, 0.0))\n r₂ = transform(Quaternion(0.0, 1.0, 0.0, 0.0))\n r₃ = transform(Quaternion(0.0, 0.0, 1.0, 0.0))\n r₄ = transform(Quaternion(0.0, 0.0, 0.0, 1.0))\n _M = reshape([vec(r₁); vec(r₂); vec(r₃); vec(r₄)], (4, 4))\n decomposition = LinearAlgebra.eigen(_M)\n λ = decomposition.values\n Λ = [λ[1] 0.0 0.0 0.0; 0.0 λ[2] 0.0 0.0; 0.0 0.0 λ[3] 0.0; 0.0 0.0 0.0 λ[4]]\n M = real.(decomposition.vectors * Λ * LinearAlgebra.inv(decomposition.vectors))","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Next, we instantiate another spin-vector using M * u = v in order to examine the effect of the transformation M on Argand plane. Specifically, the point ζ from the Argand plane of u transforms into α * s.ζ + β, where α determines the extension of Argand plane and β the translation. The scalar a controls the translation of the plane because β is defined as β = Complex(im * a). We assert that the transformation induced on Argand plane is correct by comparing the approximate equality of the Argand plane of v and the Argand plane of u. Similar to previous animation stages, the induced transformation on Argand plane by M is completely characterized using three different points: u₁, u₂, u₃. After transforming u by M we assert that the result v is still a null vector.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" u₁ = 𝕍(1.0, 1.0, 0.0, 0.0)\n u₂ = 𝕍(1.0, 0.0, 1.0, 0.0)\n u₃ = 𝕍(1.0, 0.0, 0.0, 1.0)\n for u in [u₁, u₂, u₃]\n v = 𝕍(vec(M * Quaternion(u.a)))\n @assert(isnull(v, atol = tolerance), \"v ∈ 𝕍 in not a null vector, $v.\")\n s = SpinVector(u) # TODO: visualize the spin-vectors as frames on S⁺\n s′ = SpinVector(v)\n β = Complex(im * a)\n α = 1.0\n ζ = α * s.ζ + β\n ζ′ = s′.ζ\n if ζ′ == Inf\n ζ = real(ζ)\n end\n @assert(isapprox(ζ, ζ′, atol = tolerance), \"The transformation induced on Argand plane is not correct, $ζ != $ζ′.\")\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Finally, we also assert that the null direction z + t is invariant under the transformation M because it is a null rotation with a fixed null direction at the north pole. The animation of a null rotation is correct if all of the assertions evaluate true.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" v₁ = 𝕍(normalize(ℝ⁴(1.0, 0.0, 0.0, 1.0)))\n v₂ = 𝕍(vec(M * Quaternion(vec(v₁))))\n @assert(isnull(v₁, atol = tolerance), \"vector t + z in not null, $v₁.\")\n @assert(isapprox(v₁, v₂, atol = tolerance), \"The null vector t + z is not invariant under the null rotation, $v₁ != $v₂.\")","category":"page"},{"location":"hopffibration.html#Update-the-Camera","page":"Hopf Fibration","title":"Update the Camera","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The 3D camera of the scene requires the eye position, look at, and up vectors for positioning and orientation. The function update_cam! takes the scene object along with the three required vectors as arguments and updates the camera. But, our camera position and orientation vectors are of type ℝ³, and not Vec3f. To match the argument type we need to use the generic function vec and the splat operator in order to instantiate objects of type Vec3f, because update_cam! is going to match the given type with its own signature.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" GLMakie.update_cam!(lscene.scene, GLMakie.Vec3f(vec(eyeposition)...), GLMakie.Vec3f(vec(lookat)...), GLMakie.Vec3f(vec(up)...))","category":"page"},{"location":"hopffibration.html#Record-an-Animation","page":"Hopf Fibration","title":"Record an Animation","text":"","category":"section"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Updating the base maps requires a base point in the section denoted by q and the transformation M. Then, we use M to update base maps 1, 2, 3 and 4. For we want to have different choices of an inertial reference frame in the tangent space of some point in spacetime. The generic function update! updates base maps by changing the structurally embedded observables, and then the graphical shapes take different forms accordingly.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Although we are talking about points in the bundle, embedded in ℝ⁴ and of type Quaternion, the generic function project converts them to points in ℝ³. The one method of project takes the given point q ∈ S³ ⊂ ℂ² and turns it into a point in the Euclidean space E³ ⊂ ℝ³ using stereographic projection. We identify mathbbR^4 to mathbbC^2 given by (x_1 x_2 x_3 x_4) mapsto (x_1 + i x_2 x_3 + i x_4). Then, the stereographic projection is given by: project S^3 setminus (1 0) to mathbbR^3 given by (x_1 x_2 x_3 x_4) mapsto fracx_2 x_3 x_4^T1 - x_1.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Whenever we call the update! function with an object like basemap1, giving transformation M, two things happn under the hood for deforming the graphics (update!(basemap1, q, gauge1, M)). First, a matrix of type ℝ³ is made, Matrix{ℝ³}. That is the job of one of the methods of the generic function make. The correct dispatch is selected automatically for the job, based on the argument signature (whether the first argument is of type Whirl or Basemap for example). The selected method makes a 2-surface (lattice) of the horizontal section at base point q after transforming by M, with the given segments number, gauge and chart. A chart and a gauge play the role of a choice of local trivialization of the Hopf bundle, as an atlas, for the purpose of constructing a pullback of the Earth's surface.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Second, the matrix of ℝ³ along with the given basemap's observables are passed to the function updatesurface! for updating the observables. For each coordinate component x, y and z in the Euclidean 3-space E^3, there is a corresponding matrix of real numbers, of the same size: (segments by segments). In the type structure of a Basemap or a Whirl there is a tuple whose elements are of type Observable. Each element of the three-tuple in turn contains a matrix of components x, y or z. Reshaping a matrix of 3-vectors into three matrices of scalars is done because when we implicitly instantiated a GLMakie surface in the beginning, we supplied it with three observables representing x, y and z coordinates separately. The generic function buildsurface from the source file surface.jl builds a surface with the given scene, value, color and transparency. Here, the value argument is of type Matrix{ℝ³}. The interface between the construction of our base maps (or whirls) and the graphics engine is essentially a reshaping and type conversion. See surface_tests.jl for use cases.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"Every time we update the observables of a Whirl under transformation by M, we need to access the coordinates of the boundary data (update!(whirls1[i], points[i], gauge1, gauge2, M)). But the coordinates are not changed, and instead the change-of-basis is taken care of by the map M. The coordinate component ϕ is divided by a factor of four since in geographic coordinates longitudes range from -π to +π, whereas latitudes range from -π / 2 to +π / 2 (exp(ϕ / 4 * K(1) + θ / 2 * K(2)))). This division rescales the longitude component of coordinates and allows us to have a square bundle chart, compared to coordinate components θ. Rescaling θ and ϕ aligns the boundaries of horizontal and vertical subspaces. We finish the animation of one time-step after updating the last Whirl.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"The function animate takes as input an integer called frame and updates the scene observables according to the stages that we described earlier. First, it calculates the progress of the animation frames, dividing frame by frames_number. For different properties of Lorentz transformations we have four stages, each stage having its own progress. The signature of the four-screw animator function is compute_fourscrew(progress::Float64, status::Int). For example, stage one animates a proper rotation of Argand plane by calling the function compute_fourscrew with status equal to 1. Stage 2 animates a pure z-boost. Then, stage 3 animates a four-screw. Finally, stage 4 animates a null rotation by calling the function compute_nullrotation. After calling each stage function, we update the camera by calling the function updatecamera.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" animate(frame::Int) = begin\n progress = frame / frames_number\n stage = min(totalstages - 1, Int(floor(totalstages * progress))) + 1\n stageprogress = totalstages * (progress - (stage - 1) * 1.0 / totalstages)\n println(\"Frame: $frame, Stage: $stage, Total Stages: $totalstages, Progress: $stageprogress\")\n if stage == 1\n M = compute_fourscrew(stageprogress, 1)\n elseif stage == 2\n M = compute_fourscrew(stageprogress, 2)\n elseif stage == 3\n M = compute_fourscrew(stageprogress, 3)\n elseif stage == 4\n M = compute_nullrotation(stageprogress)\n end\n update!(basemap1, q, gauge1, M)\n update!(basemap2, q, gauge2, M)\n update!(basemap3, q, gauge3, M)\n update!(basemap4, q, gauge4, M)\n for i in eachindex(whirls1)\n update!(whirls1[i], points[i], gauge1, gauge2, M)\n update!(whirls2[i], points[i], gauge2, gauge3, M)\n update!(whirls3[i], points[i], gauge3, gauge4, M)\n update!(whirls4[i], points[i], gauge4, gauge5, M)\n end\n updatecamera()\n end","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":"To create an animation you need to use the record function. In summary, we instantiated a Scene inside a Figure. Next, we created and animated observables in the scene, on a frame by frame basis. Now, we record the scene by passing the figure fig, the file path of the resulting video, and the range of frame numbers to the record function. The frame is incremented by record and the frame number is passed to the function write to animate the observables. Once the frame number reaches the total number of animation frames, recording is finished and a video file is saved on the hard drive at the file path: gallery/planethopf.mp4.","category":"page"},{"location":"hopffibration.html","page":"Hopf Fibration","title":"Hopf Fibration","text":" GLMakie.record(fig, joinpath(\"gallery\", \"$modelname.mp4\"), 1:frames_number) do frame\n animate(frame)\n end","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Description = \"News Report\"","category":"page"},{"location":"newsreport.html#Lede","page":"News Report","title":"Lede","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Show a small piece of the story.","category":"page"},{"location":"newsreport.html#Context","page":"News Report","title":"Context","text":"","category":"section"},{"location":"newsreport.html#Where,-Who,-What,-How-and-Why","page":"News Report","title":"Where, Who, What, How and Why","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Tell us the facts of the story.","category":"page"},{"location":"newsreport.html#Where","page":"News Report","title":"Where","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: fourscrew1)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"What is the relation between a spin-frame and a Minkowski tetrad? The spin-frame is denoted by omicron (black) and iota (silver). Omicron and iota serve as two flag poles, where we also show their respective flags. In order to see the flags, find the arcs in the x direction that move with omicron and iota during a series of transformations. The spin-frame is in a vector space over complex numbers. The spin space has the axioms of an abstract vector space. But, we have defined a special inner product for 2-spinors, such that the product of omicron and iota equals unity, whereas the product of iota and omicron equals minus unity. In other words, the inner product eats a pair of spin-vectors in the Hopf bundle and spits out a complex number (a scalar).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: fourscrew2)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The axes t (red), x (green), y (blue), and z (orange) are parts of a Minkowski tetrad in Minkowski spacetime. Choosing the default Minkowski tetrad, the tetrad aligns with the Cartesian axes of real dimension four. But, when we apply a spin-transformation, the tetrad no longer aligns with Cartesian coordinates, and with it the spin-frame bases omicron and iota change as well. The kinds of spin transformation that we apply are four-screws and null rotations, and so they are restricted transformations. Restricted transformations do not alter the sign of time. Here, the time sign is negative one, which is the same as the wall clock time.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: fourscrew3)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"If you look closely, there are two spheres in the middle that change hue over time. One of them is the past null cone and the other is the sphere S^{-1}. You will recognize the null cone as soon as it turns into a cone momentarily. If a spin-vector is in S^{-1}, then under restricted spin-transformations it does not leave the sphere S^{-1} to S^{+1}. The past null cone is the directions of light that reach our eyes from the past. But, the sphere S^{-1} is the light that we can observe around us in the present moment (assume we’re in deep space and away from heavenly objects). Under spin-transformations the null cone and the sphere S^{-1} change too, because they are embedded in Minkowski spacetime of dimension 4.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: nullrotation)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"timesign = -1\nο = SpinVector([Complex(1.0); Complex(0.0)], timesign)\nι = SpinVector([Complex(0.0); Complex(1.0)], timesign)\n@assert(isapprox(dot(ο, ι), 1.0), \"The inner product of spin vectors $ι and $ο is not unity.\")\n@assert(isapprox(dot(ι, ο), -1.0), \"The inner product of spin vectors $ι and $ο is not unity.\")","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"generate() = 2rand() - 1 + im * (2rand() - 1)\nκ = SpinVector(generate(), generate(), timesign)\nϵ = 0.01\nζ = Complex(κ)\nζ′ = ζ - 1.0 / √2 * ϵ / κ.a[2]\nκ = SpinVector(ζ, timesign)\nκ′ = SpinVector(ζ′, timesign)\nω = SpinVector(generate(), generate(), timesign)\nζ = Complex(ω)\nζ′ = ζ - 1.0 / √2 * ϵ / ω.a[2]\nω = SpinVector(ζ, timesign)\nω′ = SpinVector(ζ′, timesign)\n@assert(isapprox(dot(κ, ι), vec(κ)[1]), \"The first component of the spin vector $κ is not equal to the inner product of $κ and $ι.\")\n@assert(isapprox(dot(κ, ο), -vec(κ)[2]), \"The second component of the spin vector $κ is not equal to minus the inner product of $κ and $ο.\")\n@assert(isapprox(dot(ω, ι), vec(ω)[1]), \"The first component of the spin vector $ω is not equal to the inner product of $ω and $ι.\")\n@assert(isapprox(dot(ω, ο), -vec(ω)[2]), \"The second component of the spin vector $ω is not equal to minus the inner product of $ω and $ο.\")","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"t = 𝕍(1.0, 0.0, 0.0, 0.0)\nx = 𝕍(0.0, 1.0, 0.0, 0.0)\ny = 𝕍(0.0, 0.0, 1.0, 0.0)\nz = 𝕍(0.0, 0.0, 0.0, 1.0)\nο = √2 * (t + z)\nι = √2 * (t - z)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct360)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The phase of the inner product of spin-vectors is shown as a prism arc. In a Minkowski tetrad with bases t, x, y and z, (with signature (+,-,-,-)) there are a pair of basis vectors for spin-vectors: omicron and iota. For example, the spin-vectors kappa and omega, each are linear combinations of omicron and iota. The product of kappa and omega is a complex number that has a magnitude and a phase. Being spin-vectors, the arrows of omicron, iota, kappa and omega represent the flagpoles, and the flag planes are attached to the flagpoles as arcs.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct720)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In order to find the inner product of kappa and omega we make use of both flagpoles and flag planes. First, note that the flagpoles span a 2-plane in the Minkowski vector space. Then, we perform the Gram-Schmidt orthogonalization method to find the orthogonal complement of the 2-plane. Next, find the intersection of the flag planes and the orthogonal complement 2-plane from the previous step. By this step, the flag plane of kappa results in vector U, whereas the flag plane of omega projects to arrow V. Then, we normalize U and V. Finally, the angle that U and V make with each other measure pi plus two times the argument of the inner product of kappa and omega.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct1080)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Furthermore, the arrow that is denoted by p bisects the angle between U and V, and measures the phase angle minus pi half (modulus two pi). Also, a spatial rotation about the axis p is done for animating the Minkowski vector space so that all of the components of the inner product are visible from a 720-degree view.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct1440)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"A spin-vector is named kappa and another spin-vector is named omega. The extra piece of information that makes spinors special is the flagpoles of spin-vectors. Using a differential operator in the plane of complex numbers, starting with zeta complex, the spin counterpart of the spin vector zeta prime equals zeta minus one over the square root of two times a constant named epsilon, over eta (the second component of the spin-vector). Except for this transformation of zeta to zeta prime, which is parameterized by epsilon, the spin-vectors kappa and kappa prime have the same features such as time sign. The same transformation produces the names omega and omega prime. With iota and omicron as the basis vectors of the spin-space G dot, we assert the following propositions:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The first component of the spin vector κ is equal to the inner product of κ and ι.\nThe second component of the spin vector κ is equal to minus the inner product of κ and ο.\nThe first component of the spin vector ω is equal to the inner product of ω and ι.\nThe second component of the spin vector ω is equal to minus the inner product of ω and ο.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproduct)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"κ = 𝕍(κ)\nκ′ = 𝕍(κ′)\nω = 𝕍(ω)\nω′ = 𝕍(ω′)\nzero = 𝕍(0.0, 0.0, 0.0, 0.0)\nB = stack([vec(κ), vec(ω), vec(zero), vec(zero)])\nN = LinearAlgebra.nullspace(B)\na = 𝕍(N[begin:end, 1])\nb = 𝕍(N[begin:end, 2])\na = 𝕍(LinearAlgebra.normalize(vec(a - κ - ω)))\nb = 𝕍(LinearAlgebra.normalize(vec(b - κ - ω)))","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"When we stack the Minkowski vector space representation of kappa and omega and fill the rest with zero to get a square matrix B, the null space of B is where the piece of information about spinors exist. By performing a Gram-Schmidt procedure we find the set of orthonormal basis vectors for the inner product of kappa and omega. In the following lines, the spin-vectors an and b are bases of the null space of matrix B.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"v₁ = κ.a\nv₂ = ω.a\nv₃ = a.a\nv₄ = b.a\n\ne₁ = v₁\nê₁ = normalize(e₁)\ne₂ = v₂ - dot(ê₁, v₂) * ê₁\nê₂ = normalize(e₂)\ne₃ = v₃ - dot(ê₁, v₃) * ê₁ - dot(ê₂, v₃) * ê₂\nê₃ = normalize(e₃)\ne₄ = v₄ - dot(ê₁, v₄) * ê₁ - dot(ê₂, v₄) * ê₂ - dot(ê₃, v₄) * ê₃\nê₄ = normalize(e₄)\n\nê₁ = 𝕍(ê₁)\nê₂ = 𝕍(ê₂)\nê₃ = 𝕍(ê₃)\nê₄ = 𝕍(ê₄)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproductspositiveus)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The flag planes of kappa and omega are obtained by subtracting kappa from kappa prime and omega from omega prime, respectively. Projecting the flag plane of kappa onto the 2-plane spanned by subspace bases of ê₃ and ê₄ gives you vector U. The same subspace gives you V for the flag plane of omega. The inner product eats two spin-vectors such as kappa and omega, and spits out a complex number that has a magnitude and a phase angle. The angle that U and V make with each other determines the phase of the inner product times two plus pi. This 2-plane is the orthogonal complement of the 2-plane that contains kappa and omega (and is spanned by ê₁ and ê₂). The camera looks at the sum of the vectors kappa and omega.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"κflagplanedirection = 𝕍(LinearAlgebra.normalize(vec(κ′ - κ)))\nωflagplanedirection = 𝕍(LinearAlgebra.normalize(vec(ω′ - ω)))\nglobal u = LinearAlgebra.normalize(vec((-dot(ê₃, κflagplanedirection) * ê₃ + -dot(ê₄, κflagplanedirection) * ê₄)))\nglobal v = LinearAlgebra.normalize(vec((-dot(ê₃, ωflagplanedirection) * ê₃ + -dot(ê₄, ωflagplanedirection) * ê₄)))\np = 𝕍(LinearAlgebra.normalize(u + v))\nglobal p = -dot(ê₃, p) * ê₃ + -dot(ê₄, p) * ê₄\naxis = normalize(ℝ³(vec(p)[2:4]))\nM = mat4(Quaternion(progress * 4π, axis))\nο_transformed = M * Quaternion(vec(ο))\nι_transformed = M * Quaternion(vec(ι))","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: innerproductspositivechina)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"ζ = Complex(_κ + _ω)\n_τ = SpinVector(ζ, timesign)\nζ′ = Complex(_κ′ + _ω′)\n_τ′ = SpinVector(ζ′, timesign)\ngauge1 = -imag(dot(_κ, _ω))\ngauge2 = -imag(dot(_κ, _τ))\ngauge3 = float(π)\n@assert(isapprox(dot(_τ, _ι), vec(_τ)[1]), \"The second component of the spin vector $_τ is not equal to minus the inner product of $_τ and $_ι.\")\n@assert(isapprox(dot(_τ, _ο), -vec(_τ)[2]), \"The second component of the spin vector $_τ is not equal to minus the inner product of $_τ and $_ο.\")","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The geometry of \"spin-vector addition\" is shown. The spin-vectors exist in a spin-space that is equipped with three operations: scalar multiplication, inner product and addition. The addition of spin-vectors κ and ω results in another spin-vector κ + ω in the spin-space, which has its own flagpole and flag plane. Taking κ and ω as null vectors in the sphere of future null directions, the flagpole of κ is represented by a point (complex number) and the null flag of κ is represented as a point sufficiently close to κ that is used to assign a direction tangent to the sphere at κ.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: addition02)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The tails of the flagpoles of κ, ω and κ + ω are in a circle in the sphere of future null directions. The circumcircle of the triangle made by joining the tails of the three spin-vectors makes angles with the flagpoles and null planes. Meaning, the distance between κ and the center of the circle is equal to the distance between ω and the center. Also, the distance of the addition of κ and ω and the circle center is the same as the distance between κ and the center. For the circumcircle, we have three collinear points in the Argand complex plane. However, lines in the Argand plane become circles in sections of the three-dimensional sphere. The angle that the flagpoles of κ and ω make with the circle should be twice the argument of the inner product of the two spin-vectors (modulus 2π with a possible addition of π).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: addition08)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"w = (Complex(κ + ω) - Complex(κ)) / (Complex(ω) - Complex(κ))\n@assert(imag(w) ≤ 0 || isapprox(imag(w), 0.0), \"The flagpoles are not collinear: $(Complex(κ)), $(Complex(ω)), $(Complex(κ + ω))\")","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In an interesting way, the argument (phase) of the inner product of κ and ω is equal to half of the sum of the angles that the spin-vectors make with the circle, which is in turn equal to the angle that U and V make with each other minus π (also see the geometric descriptions of the inner product to construct U and V). In the case of spin-vector addition, the angles that the flag planes of κ, ω and κ + ω, each make with the circle are equal. But, be careful with determining the signs of the flag planes and the possible addition of π to the flag plane of κ + ω. For determining flag plane signs, see also Figure 1-21 in page 64 of Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: addition09)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"For example, the Standard Model is formulated on 4-dimensional Minkowski spacetime, over which all fiber bundles can be trivialized and spinors have a simple explicit description. For the Symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In theoretical physics, Lie groups like the Lorentz and Poincaré groups, which are related to spacetime symmetries, and gauge groups, defining internal symmetries, are important cornerstones. Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group. The Lie algebra SL(2mathbbC) plays a special role in physics, because as a real Lie algebra it is isomorphic to the Lie algebra of the Lorentz group of 4-dimensional spacetime. At least locally, fields in physics can be described by maps on spacetime with values in vector spaces.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. As we will discuss in Sect. 6.8.2 in more detail, the group SL(2mathbbC) is the (orthochronous) Lorentz spin group, i.e. the universal covering of the identity component of the Lorentz group of 4-dimensional spacetime. The fundamental geometric opbject in a gauge theory is a principal bundle over spacetime with structure group given by the gauge group. The fibers of a principal bundle are sometimes thought of as an internal space at every spacetime point, not belonging to spacetime itself. Fiber bundles are indispensible in gauge theory and physics in the situation where spacetime, the base manifold, has a non-trivial topology.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It also happens if we compactify (Euclidean) spacetime mathbbR^4 to the 4-sphere S^4. In these situations, fields on spacetime often cannot be described simply by a map to a fixed vector space, but rather as sections of a non-trivial vector bundle. We will see that this is similar to the difference in special relativity between Minkowski spacetime and the choice of an inertial system. This can be compared, in special relativity, to the choice of an inertial system for Minkowski spacetime M, which defines an identification on M cong mathbbR^4. Of course, different choices of gauges are possible, leading to different trivializations of the principal bundle, just as different choices of inertial systems lead to different identifications of spacetime with mathbbR^4.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Note that, if we consider principal bundles over Minkowski spacetime mathbbR^4, it does not matter for this discussion that principal bundles over Euclidean spaces are always trivial by Corollary 4.2.9. This is very similar to special relativity, where spacetime is trivial, i.e. isomorphic to mathbbR^4 with a Minkowski metric, but what matters is the independence of the actual trivialization, i.e. the choice of inertial system. Table 4.2 Comparison between notions for special relativity and gauge theory","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":" Manifold Trivialization Transformations and invariance\nSpecial relativity Spacetime M M cong mathbbR^4 via inertial system Lorentz\nGauge theory Principal bundle P to M P cong M times G via choice of gauge Gauge","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It follows that, given a local gauge of the gauge bundle P, the section in E corresponds to a unique local map from spacetime into the vector space V. In particular, we can describe matter fields on a spacetime diffeomorphic to mathbbR^4 by unique maps from mathbbR^4 into a vector space, once a global gauge for the principal bundle has been chosen. At least locally (after a choice of local gauge) we can interpret connection 1-forms as fields on spacetime (the base manifold) with values in the Lie algebra of the gauge group. Notice that connections are not unique (if dim M dim G ge 1), not even in the case of trivial principal bundles (all connections that appear in the Standard Model over Minkowski spacetime, for example, are defined on trivial principal bundles). The diffeomorphism group Diff(M) of spacetime M plays a comparable role in general relativity.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"This is related to the fact that gauge theories describe local interactions (the interactions occur in single spacetime points). The local connection 1-form is thus defined on an open subset in the base manifold M and can be considered as a \"field on spacetime\" in the usual sense. Generalized Electric and Magnetic Fields on Minkowski Spacetime of Dimension 4 In quantum field theory, the gauge field A_mu is a function on spacetime with values in the operators on the Hilbert state space V (if we ignore for the moment questions of whether this operator is well-defined and issues of regularization). By Corollary 5.13.5 this difference can be identified with a 1-form on spacetime M with values in Ad(P).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In physics this fact is expressed by saying that gauge bosons, the differences A_mu-A_mu^0, are fields on spacetime that transform in the adjoint representation of G under gauge transformations. In the case of Minkowski spacetime, rotations correspond to Lorentz transformations. The pseudo-Riemannian case, like the case of Minkowski spacetime, is discussed less often, even though it is very important for physics (a notable exception is the thorough discussion in Helga Baun's book [13]). mathbbR^s1 and mathbbR^1t are the two versions of Minkowski spacetime (both versions are used in physics). This includes the particular case of the Lorentz group of Minkowski spacetime.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"However, as mentioned above, depending on the convention, 4-dimensional Minkowski spacetime in quantum field theory can have signature (+---), so that time carries the plus sign. Example 6.1.20 For applications concerning the Standard Model, the most important of these groups is the proper orthochronous Lorentz group SO^+(13) cong SO^+(31) of 4-dimensional Minkoeski spacetime. They are physical gamma matrices for Cl(13), i.e. for the Clifford algebra of Minkowski spacetime with signature (+---), in the so-called Weyl representation or chiral representation. Example 6.3.18 Let Gamma_a and gamma_a = i Gamma_a be the physical and mathematical gamma matrices for Cl(13) considered in Example 6.3.17. If we set Gamma_a^prime = gamma_a, gamma_a^prime = i Gamma_a^prime = -Gamma_a, then these are physical and Mathematical gamma matrices for Cl(13) of Minkowski spacetime with signature (-+++). Example 6.3.24 For Minkowski spacetime of dimension 4 we have Table 6.1 Complex Clifford algebras","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"n Cl(n) Cl^0(n) N\nEvan End(mathbbC^N) End(mathbbC^N2) oplus End(mathbbC^N2) 2^n2\nOdd End(mathbbC^N) oplus End(mathbbC^N) End(mathbbC^N) 2^(n-1)2","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Table 6.2 Real Clifford algebras","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"rho mod 8 Cl(st) N\n0 End(mathbbR^N) 2^n2\n1 End(mathbbC^N) 2^(n-1)2\n2 End(mathbbH^N) 2^(n-2)2\n3 End(mathbbH^N) oplus End(mathbbH^N) 2^(n-3)2\n4 End(mathbbH^N) 2^(n-2)2\n5 End(mathbbC^N) 2^(n-1)2\n6 End(mathbbR) 2^n2\n7 End(mathbbR^N) oplus End(mathbbR^N) 2^(n-1)2","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Table 6.3 Even part of real Clifford algebras","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"rho mod 8 Cl^0(st) N\n0 End(mathbbR^N) oplus End(mathbbR^N) 2^(n-2)2\n1 End(mathbbR^N) 2^(n-1)2\n2 End(mathbbC^N) 2^(n-2)2\n3 End(mathbbH^N) 2^(n-3)2\n4 End(mathbbH^N) oplus End(mathbbH^N) 2^(n-4)2\n5 End(mathbbH^N) 2^(n-3)2\n6 End(mathbbC^N) 2^(n-2)2\n7 End(mathbbR^N) 2^(n-1)2","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Cl(13) cong End(mathbbR^4)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Cl(31) cong End(mathbbH^2)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Cl^0(13) cong Cl^0(31) cong End(mathbbC^2)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Example 6.6.7 For Minkowski spacetime mathbbR^n-11 of dimension n we have n = rho + 2.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"We see that in Minkowski spacetime of dimension 4 there exist both Majorana and Weyl spinors of real dimension 4, but not Majorana-Weyl spinors. In quantum field theory, spinors become fields of operators on spacetime acting on a Hilbert space. Explicit formulas for Minkowski Spacetime of Dimension 4 We collect some explicit formulas concerning Clifford algebras and spinors for the case of 4-dimensional Minkowski spacetime. In Minkowski spacetime of dimension 4 and signature (+---) (usually used in quantum field theory) there exist both Weyl and Majorana spinors, but not Majorana-Weyl spinors. Our aim in this subsection is to prove that the orthochronous spin group Spin^+(13) of 4-dimensional Minkowski spacetime is isomorphic to the 6-dimensional Lie group SL(2mathbbC).","category":"page"},{"location":"newsreport.html#The-Story","page":"News Report","title":"The Story","text":"","category":"section"},{"location":"newsreport.html#Who","page":"News Report","title":"Who","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"With the discovery of a new particle, announced on 4 July 2012 at CERN, whose properties are \"consistent with the long-sought Higgs boson\" [31], the final elementary particle predicted by the classical Standard Model of particle physics has been found. Interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs bosons), determined by the Lagrangian. The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. The fact that there are 8 gluons, 3 weak gauge bosons, and 1 photon is related to the dimensions of the Lie groups SU(3) and SU(2) times U(1). Lie algebras are also important in gauge theories: connections on principal bundles, also known as gauge boson fields, are (locally) 1-forms on spacetime with values in the Lie algebra of the gauge group.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The adjoint representation is also important in physics, because gauge bosons correspond to fields on spacetime that transform under the adjoint representation of the gauge group. We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields. The gauge bosons corresponding to these gauge groups are described by the adjoint representation that we discuss in Sect. 2.1.5. The representation Ad_H describes the representation of the gauge boson fields in the Standard Model. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Connections on principal bundles, that we discuss in Chap. 5, correspond to gauge fields whose particle excitations in the associated quantum field theory are the gauge bosons that transmit interactions. These fields are often called gauge fields and correspond in the associated quatum field theory to gauge bosons. This implies a direct interaction between gauge bosons (the gluons in QCD) that does not occur in abelian gauge theories like quantum electrodynamics (QED). The difficulties that are still present nowadays in trying to understand the quantum version of non-abelian gauge theories, like quantum chromodynamics, can ultimately be traced back to this interaction between gauge bosons. The real-valued fields A_mu^a in C^infty(UmathbbR) and the corresponding real-valued 1-forms A_s in Omega^1(U) are called (local) gauge boson fields.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In physics, the quadratic term A_mu A_nu in the expression for F_munu (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field A_mu. This explains why gluons, the gauge bosons of QCD, interact directly with each other, while photons, the gauge bosons of QED, do not. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by A_mu and the particles described by the field phi. We then get a better understanding of why gauge bosons in physics are said to transform under the adjoint representation.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Strictly speaking, gauge bosons, the excitations of the gauge field, should then be described classically by the difference A - A^0, where A is some other connection 1-form and not by the field A itself. In physics this fact is expressed by saying that gauge bosons, the differences A_mu - A_mu^0, are fields on spacetime that transform in the adjoint represntation of G under gauge transformations. Gauge fields correspond to gauge bosons (spin 1 particles) and are described by 1-forms or, dually, vector fields. Even though spinors are elementary objects, some of their properties (like the periodicity modulo 8, real and quaternionic structures, or bilinear and Hamiltonian scalar products) are not at all obvious, already on the level of linear algebra, and do not have a direct analogue in the bosonic world of vectors and tensors. The existence of gauge symmetries is particularly important: it can be shown that a quantum field theory involving massless spin 1 bosons can be consistent (i.e. unitary, see Sect. 7.1.3) only if it is gauge invariant [125,143].","category":"page"},{"location":"newsreport.html#Graph","page":"News Report","title":"Graph","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: graph)","category":"page"},{"location":"newsreport.html#What","page":"News Report","title":"What","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The Higgs mechanism of mass generation for gauge bosons as well as the mass generation for fermions via Yukawa couplings. Spin groups such as the universal covering of the Lorentz group and its higher dimesnional analogues, are also important in physics, because they are involved in the mathematical description of fermions. Counting in this way, the Standard Model thus contains at the most elementary level 90 fermions (particles and antiparticles). The complex vector space V of fermions, which carries a representation of G, has dimension 45 (plus the same number of corresponding antiparticles) and is the direct sum of the two G-invariant subspaces (sectors): a lepton sector of dimension 9 (where we do not include the hypothetical right-handed neutrinos) and a quark sector of dimension 36. Matter fields in the Standard Model, like quarks and leptons, or sacalar fields, like the Higgs field, correspond to sections of vector bundles associated to the principal bundle (and twisted by spinor bundles in the case of fermions).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"For example, in the Standard Model, one generation of fermions is described by associated complex vector bundles of rank 8 for left-handed fermions and rank 7 for right-handed fermions, associated to representations of the gauge group SU(3) times SU(2) times U(1). Matter fields in physics are described by smooth sections of vector bundles E associated to principal bundles P via the representations of the gauge group G on a vector space V (in the case of fermions the associated bundle E is twisted in addition with a spinor bundle S, i.e. the bundle is S otimes E). Additional matter fields, like fermions or scalars, can be introduced using associated vector bundles. These particles are fermions (spin frac12 particles) and are described by spinor fields (spinors). Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly neutrinos) and, together with the Dirac operator, the kinetic term and the interaction term; see Sect. 7.6. This is related to the fact that the weak interaction in the Standard Model is not invariant under parity inversion that exchanges left-handed with right-handed fermions.","category":"page"},{"location":"newsreport.html#Perspective","page":"News Report","title":"Perspective","text":"","category":"section"},{"location":"newsreport.html#How","page":"News Report","title":"How","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Hence, by the uniqueness of integral curves (which is a theorem about the uniqueness of solutions to odrinary differential equations) we have phi_X(s) cdot phi_X(t) = phi_X(s + t) forall t in I cap (t_min - s t_max - s). This implies the claim by uniqueness of solutions of ordinary differential equations. The unique solution of this differential equation for gamma(t) is gamma(t) = e^tr(X)t. Then e^D = beginbmatrix e^d_1 0 0 0 e^d_2 0 ddots 0 0 e^d_n endbmatrix and the equation det(e^D) = e^d_1 e^d_n = e^d_1 + + d_n = e^tr(D) is trivially satisfied. Then we can calculate: (R^*_gs)_p(XY) = L_(pg)^-1*R_g*(X) L_(pg)^-1*R_g*(Y) = Ad_g^-1 circ L_p^-1*(X) Ad_g^-1 circ L_p^-1*(Y) and s_p(XY) = L_p^-1*(X) L_p^-1*(Y), where in both equations we used that s is left invariant.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Lemma 3.3.3 For A in Mat(m times m mathbbH) and v in mathbbH^m the following equation holds: detbeginbmatrix1 v 0 Aendbmatrix = det(A). Lemma 4.1.13 (Cocycle Conditions) The transition functions phi_ij_ij in I satisfy the following equations:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"phi_ii(x) = Id_F for x in U_i,\nphi_ij(x) circ phi_ji(x) = Id_F for x in U_i cap U_j,\nphi_ik(x) circ phi_kj(x) circ phi_ji(x) = Id_F for x in U_i cap U_j cap U_k.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The third equation is called the cycycle condition.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"5.5.2 The structure equation Theorem 5.5.4 (Structure Equation) The curvature form F of a connection form A satisfies F = dA + frac12AA.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proof We check the formula by inserting XY in T_pP on both sides of the equation, where we distinguish the following three cases:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Both X and Y are vertical: Then X and Y are fundamental vectors, X = tildeV_p Y = tildeW_p for certain elements VW in g. We get F(XY) = dA(pi_H(X) pi_H(Y)) = 0. On the other hand we have frac12AA(XY) = A(X)A(Y) = VW. The differential dA of a 1-form A is given according to Proposition A.2.22 by dA(XY) = L_X(A(Y))-L_Y(A(X))-A(XY), where we extend the vectors X and Y to vector fields in a neighbourhood of p. If we choose the extension by fundamental vector fields tildeV and tildeW, then dA(XY) = L_X(W) - L_Y(V) - VW = -VW since V and W are constant maps from P to g and we used that tildeVtildeW = tildeVW according to Proposition 3.4.4. This implies the claim.\nBoth X and Y are horizontal: Then F(XY) = dA(XY) and frac12AA(XY) = A(X) A(Y) = 00=0. This implies the claim.\nX is vertical and Y is horizontal: Then X = tildeV_p for some V in g. We have F(XY) = dA(pi_H(X)pi_H(Y)) = dA(0 Y) = 0 and frac12AA(XY) = A(X)A(Y) - V0 = 0. Furthermore, dA(XY) = L_tildeV(A(Y)) - L_Y(V) - A(tildeVY) = -A(tildeVY) = 0 since tildeVY is horizontal by Lemma 5.5.5. This implies the claim.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The structure equation is very useful when we want to calculate the curvature of a given connection.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"By the structure equation we have F = dA + frac12 A A so that dF = frac12 dA A.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proposition 5.6.2 (Local Structure Equation) The local field strength can be calculated as F_s = dA_s + frac12A_sA_s and F_munu = partial_mu A_nu - partial_nu A_mu + A_mu A_nu.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It remains to check that F_M is closed. In a local gauge s we have according to the local structure equation F_s = dA_s + frac12A_sA_s.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proposition 5.6.8 For the connection on the Hopf bundle the following equation holds: frac12pi i int_S^2 F_S^2 = 1.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"We write A_mu = A_s(partial_mu) F_munu = F_s(partial_mu partial_nu) and we have the local structure equation F_munu = partial_mu A_nu - partial_nu A_mu + A_mu A_nu.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"We will determine g(t) as the solution of a differential equation.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proof Properties 1-3 follow from the theory of ordinary differential equations. (Parallel transport)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Recall that for the proof of Theorem 5.8.2 concerning the existence of a horizontal lift gamma^* of a curve gamma01 to M where gamma^*(0) = p in P_gamma(0), we had to solve the differential equation dotg(t) = -R_g(t)* A(dotdelta(t)), with g(0) = e, where delta is some lift of gamma and g01 to G is a map with gamma^*(t) = delta(t) cdot g(t).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Then the differential equation can be written as fracdg(t)dt = -A_s(dotgamma(t)) cdot g(t).","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Path-ordered exponentials are useful, because they define solutions to the ordinary differential equation we are interested in.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Then uniqueness of the solution to ordinary differential equations show that g equiv h, hence g takes values in G.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The solution to this differential equation is","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"g(t) = P exp(- int_0^t sum_mu=1^n A_smu(gamma(s))fracdx^mudsds) = P exp(- int_gamma(0)^gamma(t) sum_mu=1^n A_smu (x^mu) dx^mu) = P exp(- int_gamma_t A_s), where gamma_t denotes the restriction of the curve gamma to 0t.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"What is the interpretation of the structure equation?","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Taking the determinant of both sides of this equation shows that:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Lemma 6.1.7 Matrices A in O(st) satisfy detA = pm 1.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"A^T beginbmatrix I_s 0 0 -I_t endbmatrix A = beginbmatrix I_s 0 0 -I_t endbmatrix.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Remark 6.2.5 We can think of the linear map gamma as a linear square root of the symmetric bilinear form -Q: in the definition of Clifford algebras, it suffices to demand that gamma(v)^2 = -Q(vv) cdot 1 forall v w in V, because, considering this equation for vectors v w v + w, the equation gamma(v) gamma(w = -2Q(v w) cdot 1 forall v w in V follows.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Lemma 6.3.6 Every chirality element omega satisfies","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"omegae_a = 0\nomegae_a cdot e_b = 0 forall 1 le a b le n.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Proof The first equation follows from e_a cdot omega = lambda e_a cdot e_1 e_n = (-1)^a - 1 lambda e_1 e_a cdot e_a e_n","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"omega cdot e_a = lambda e_1 e_n cdot e_a = (-1)^n - a lambda e_1 e_a cdot e_a e_n = -e_a cdot omega,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"since n is even. The second equation is a consequence of the first.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Let Gamma_1 Gamma_n be physical gamma matrices. We set","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma_a = eta^ac Gamma_c,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^bc = frac12 Gamma^b Gamma^c = frac12 (Gamma^b Gamma^c - Gamma^c Gamma^b),","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^n + 1 = -i^k + t Gamma^1 Gamma^n","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"and similarly for the mathematical gamma-matrices (in the first equation there is an implicit sum over c; this is an instance of the Einstein summation convention). These matrices satisfy by Lemma 6.3.6","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^n + 1 Gamma^a = 0,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Gamma^n + 1 Gamma^bc = 0,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"gamma^bc = -Gamma^bc.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In the following examples we use the Pauli matrices","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"sigma_1 = beginbmatrix 0 1 1 0 endbmatrix, sigma_2 = beginbmatrix 0 -i i 0 endbmatrix, sigma_3 = beginbmatrix 1 0 0 -1 endbmatrix.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"It is easy to check that they satisfy the identities","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"sigma^2 = I_2 j = 1 2 3,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"sigma_j sigma_j + 1 = -sigma_j + 1 sigma_j = i sigma_j + 2 j = 1 2 3,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"where in the second equation j + 1 and j + 2 are taken mod 3.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(psi phi) = psi^T C phi","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Furthermore property 1. and 2. in Definition 6.7.1 are equivalent to","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"gamma_a^T = mu C gamma_a C^-1 for all a = 1 s + t.\nC^T = nu C.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The first equation also holds with the physical Clifford matrices Gamma_a instead of the mathematical matrices gamma_a.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"There is an equivalent equation to the first one with physical Clifford matrices Gamma_a 1 cdot Gamma^dagger_a = -delta A Gamma_a A^-1 for all a = 1 s + t.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Furthermore, property 1. and 2. in Definition 6.7.8 are equivalent to:","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"gamma_a^dagger = delta A gamma_a A^-1 for all a = 1 s + t.\nA^dagger = A.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Given a spin structure on a pseudo-Riemannian manifold and the spinor bundle S, we would like to have a covariant derivative on S so that we can define field equations involving derivatives of spinors.","category":"page"},{"location":"newsreport.html#The-Iconic-Wall","page":"News Report","title":"The Iconic Wall","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: corrected-wall)","category":"page"},{"location":"newsreport.html#Tome","page":"News Report","title":"Tome","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: tome)","category":"page"},{"location":"newsreport.html#Wrap-Up","page":"News Report","title":"Wrap Up","text":"","category":"section"},{"location":"newsreport.html#Why","page":"News Report","title":"Why","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The following three chapters discuss applications in physics: the Lagrangians and interactions in the Standard Model, spontaneous symmetry breaking, the Higgs mechanism of mass generation, and some more advanced and modern topics like neutrino masses and CP violation. Depending on the time, the interests and the prior knowledge of the reader, he or she can take a shortcut and immediately start at the chapters on connections, spinors or Lagrangians, and then go back if more detailed mathematical knowledge is required at some point. An interesting and perhaps underappreciated fact is that a substantial number of phenomena in particle physics can be understood by analysing representations of Lie groups and by rewriting or rearranging Lagrangians.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Symmetries of Lagrangians interactions between fields corresponding to elementary particles (quarks, leptons, gauge bosons, Higgs boson), determined by the Lagrangian. For the symmetries relevant in field theories, the groups act on fields and leave the Lagrangian or the action (the spacetime integral over the Lagrangian) invariant. In the following chapter we will study some associated concepts, like representations (which are used to define the actions of Lie groups on fields) and invariant matrices (which are important in the construction of the gauge invariant Yang-Mills Lagrangian). We also discuss special scalar products on Lie algebras which will be used in Sect. 7.3.1 to construct Lagrangians for gauge boson fields.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"The existence of positive definite Ad-invariant scalar products on the Lie algebra of compact Lie groups is very important in gauge theory, in particular, for the construction of the gauge-invariant Yang-Mills Lagrangian; see Sect. 7.3.1. The fact that these scalar products are positive definite is important from a phenomenological point of view, because only then do the kinetic terms in the Yang-Mills Lagrangian have the right sign (the gauge bosons have positive kinetic energy [148]). In a gauge-invariant Lagrangian this results in terms of order higher than two in the matter and gauge fields, which are interpreted as interactions between the corresponding particles. In non-abelian gauge theories, like quantum chromodynamics (QCD), there are also terms in the Lagrangian of order higher than two in the gauge fields themselves, coming from a quadratic term in the curvature that appears in the Yang-Mills Lagrangian.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"In physics, the quadratic term A_mu A_nu in the expression for F_munu (leading to cubic and quartic terms in the Yang-Mills Lagrangian, see Definition 7.3.1 and the corresponding local formula in Eq. (7.1)) is interpreted as a direct interaction between gauge bosons described by the gauge field A_mu. These covariant derivatives appear in physics, in particular, in the Lagrangians and field equations defining gauge theories. This non-linearity, called minimal coupling, leads to non-quadratic terms in the Lagrangian (see Definition 7.5.5 and Definition 7.6.2 as well as the local formulas in Eqs. (7.3) and (7.4)), which are interpreted as an interaction between gauge bosons described by A_mu and the particles described by the field phi.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"(Image: feynmandiagrams)","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Figure 5.2 shows the Feynman diagrams for the cubic and quartic terms which appear in the Klein-Gordon Lagrangian in Eq. (7.3), representing the interaction between a gauge field A and a charged scalar field described locally by a map phi with values in V. Fig 5.2 Feynman diagrams for interaction between gauge field and charged scalar","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Hermitian scalar products are particularly important, because we need them in Chap. 7 to define Lorentz invariant Lagrangians involving spinors.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"psi phi = overlinepsi phi,","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"overlinepsi = psi^dagger A.","category":"page"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Dirac forms are used in the Standard Model to define a Dirac mass term in the Lagrangian for all fermions (except possibly the neutrinos) and, together with the Dirac operator the kinetic term and the interaction term; see Sect. 7.6.","category":"page"},{"location":"newsreport.html#Porta.jl","page":"News Report","title":"Porta.jl","text":"","category":"section"},{"location":"newsreport.html#References","page":"News Report","title":"References","text":"","category":"section"},{"location":"newsreport.html","page":"News Report","title":"News Report","text":"Mark J.D. Hamilton, Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics, Springer Cham, DOI, published: 10 January 2018.\nSir Roger Penrose, The Road to Reality, (2004).\nRoger Penrose, Wolfgang Rindler, Spinors and Space-Time, Volume 1: Two-spinor calculus and relativistic fields, (1984).\nRichard M. Murray and Zexiang Li, A Mathematical Introduction to Robotic Manipulation, 1st Edition, 1994, CRC Press, read, buy.\nEdward Witten, Physics and Geometry, (1987).\nThe iconic Wall of Stony Brook University.","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Description = \"How the reaction wheel unicycle works.\"","category":"page"},{"location":"reactionwheelunicycle.html#The-Reaction-Wheel-Unicycle","page":"Reaction Wheel Unicycle","title":"The Reaction Wheel Unicycle","text":"","category":"section"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_cnt = beginbmatrix dotx - r_w dottheta cos(delta) newline doty - r_w dottheta sin(delta) newline dotz endbmatrix = beginbmatrix 0 newline 0 newline 0 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"dotx = r_w dottheta cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"doty = r_w dottheta sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"dotz = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotq_i) - fracpartial Lpartial q_i = Q_i + sum_k=1^n lambda_k a_ki","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"i = 1 ldots m","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"L = T_total - P_total","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_w2^cpT = beginbmatrix 1 0 0 0 newline 0 cos(alpha) -sin(alpha) 0 newline 0 sin(alpha) cos(alpha) 0 newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 r_w newline 0 0 0 1 endbmatrix = beginbmatrix 1 0 0 0 newline 0 cos(alpha) -sin(alpha) -r_w sin(alpha) newline 0 sin(alpha) cos(alpha) r_w cos(alpha) newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_cp^gT = beginbmatrix cos(delta) -sin(delta) 0 x newline sin(delta) cos(delta) 0 y newline 0 0 1 0 newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_w2^gT = _cp^gT times _w2^cpT = beginbmatrix cos(delta) -sin(delta) cos(alpha) sin(delta) sin(alpha) x + r_w sin(delta) sin(alpha) newline sin(delta) cos(delta) cos(alpha) -cos(delta) sin(alpha) y - r_w cos(delta) sin(alpha) newline 0 sin(alpha) cos(alpha) r_w cos(alpha) newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^w2P_w = beginbmatrix 0 newline 0 newline 0 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gP_w = _w2^gT times ^w2P_w = beginbmatrix x + r_w sin(alpha) sin(delta) newline y - r_w sin(alpha) cos(delta) newline r_w cos(alpha) newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^w2T = beginbmatrix cos(beta) 0 sin(beta) 0 newline 0 1 0 0 newline -sin(beta) 0 cos(beta) 0 newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 l_c newline 0 0 0 1 endbmatrix = beginbmatrix cos(beta) 0 sin(beta) l_c sin(beta) newline 0 1 0 0 newline -sin(beta) 0 cos(beta) l_c cos(beta) newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gT = _w2^gT times _c^w2T = beginbmatrix _c^gt_11 -sin(delta) cos(alpha) _c^gt_13 _c^gt_14 newline _c^gt_21 cos(delta) cos(alpha) _c^gt_23 _c^gt_24 newline -cos(alpha) sin(beta) sin(alpha) cos(alpha) cos(beta) _c^gt_34 newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_11 = cos(beta) cos(delta) - sin(alpha) sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_13 = sin(beta) cos(delta) + sin(alpha) cos(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_14 = x + r_w sin(delta) sin(alpha) + l_c sin(beta) cos(delta) + l_c sin(alpha) cos(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_21 = cos(beta) sin(delta) + sin(alpha) sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_23 = sin(beta) sin(delta) - sin(alpha) cos(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_24 = y - r_w cos(delta) sin(alpha) + l_c sin(beta) sin(delta) - l_c sin(alpha) cos(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_c^gt_34 = r_w cos(alpha) + l_c cos(alpha) cos(beta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^cP_c = beginbmatrix 0 newline 0 newline 0 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gP_c = _c^gT times ^cP_c = beginbmatrix ^gp_c1 newline ^gp_c2 newline ^gp_c3 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_c1 = x + r_w sin(alpha) sin(delta) + l_c cos(beta) sin(alpha) sin(delta) + l_c sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_c2 = y - r_w sin(alpha) cos(delta) - l_c cos(beta) sin(alpha) cos(delta) + l_c sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_c3 = r_w cos(alpha) + l_c cos(beta) cos(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^cT = beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 l_cr newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 cos(gamma) -sin(gamma) 0 newline 0 sin(gamma) cos(gamma) 0 newline 0 0 0 1 endbmatrix beginbmatrix 1 0 0 0 newline 0 1 0 0 newline 0 0 1 0 newline 0 0 0 1 endbmatrix = beginbmatrix 1 0 0 0 newline 0 cos(gamma) -sin(gamma) 0 newline 0 sin(gamma) cos(gamma) l_cr newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gT = _c^gT times _r^cT = beginbmatrix _r^gt_11 _r^gt_12 _r^gt_13 _r^gt_14 newline _r^gt_21 _r^gt_22 _r^gt_23 _r^gt_24 newline -cos(alpha) sin(beta) _r^gt_32 _r^gt_33 _r^gt_34 newline 0 0 0 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_11 = cos(beta) cos(delta) - sin(alpha) sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_12 = -sin(delta) cos(alpha) cos(gamma) + cos(delta) sin(beta) sin(gamma) + sin(delta) sin(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_13 = sin(delta) cos(alpha) sin(gamma) + cos(delta) sin(beta) cos(gamma) + sin(delta) sin(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_14 = 0 + l_cr (cos(delta) sin(beta) + sin(delta) sin(alpha) cos(beta)) + l_c sin(beta) cos(delta) + l_c cos(beta) sin(delta) sin(alpha) + x + r_w sin(delta) sin(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_21 = cos(beta) sin(delta) + sin(alpha) sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_22 = cos(delta) cos(alpha) cos(gamma) + sin(delta) sin(beta) sin(gamma) - cos(delta) sin(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_23 = -cos(delta) cos(alpha) sin(gamma) + sin(delta) sin(beta) cos(gamma) - cos(delta) sin(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_24 = l_cr (sin(delta) sin(beta) - cos(delta) sin(alpha) cos(beta)) + l_c sin(beta) sin(delta) - l_c cos(beta) cos(delta) sin(alpha) + y - r_w cos(delta) sin(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_32 = sin(alpha) cos(gamma) + cos(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_33 = -sin(alpha) sin(gamma) + cos(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^gt_34 = l_cr cos(alpha) cos(beta) + l_c cos(beta) cos(alpha) + r_w cos(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^rP_r = beginbmatrix 0 newline 0 newline 0 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gP_r = _r^gT times ^rP_r = beginbmatrix ^gp_r1 newline ^gp_r2 newline ^gp_r3 newline 1 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_r1 = x + r_w sin(alpha) sin(delta) + (l_c + l_cr) cos(beta) sin(alpha) sin(delta) + (l_c + l_cr) sin(beta) cos(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_r2 = y - r_w sin(alpha) cos(delta) - (l_c + l_cr) cos(beta) sin(alpha) cos(delta) + (l_c + l_cr) sin(beta) sin(delta)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"^gp_r3 = r_w cos(alpha) + (l_c + l_cr) cos(beta) cos(alpha)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_w = fracdP_wdt","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_c = fracdP_cdt","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"V_r = fracdP_rdt","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Omega_w = beginbmatrix 0 newline dottheta newline 0 newline 0 endbmatrix + beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _g^w2T times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix 0 newline dottheta newline 0 newline 0 endbmatrix + beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _w2^gT^-1 times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotalpha newline dottheta + dotdelta sin(alpha) newline dotdelta cos(alpha) endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Omega_c = beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _w2^cT times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _g^cT times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _c^w2T^-1 times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _c^gT^-1 times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotalpha cos(beta) - dotdelta cos(alpha) sin(beta) newline dotbeta + dotdelta sin(alpha) newline dotalpha sin(beta) + dotdelta cos(alpha) cos(beta) newline 0 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"_r^w2T = _w2^gT^-1 times _r^gT","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Omega_r = beginbmatrix dotgamma newline 0 newline 0 newline 0 endbmatrix + _c^rT times beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _w2^rT times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _g^rT times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotgamma newline 0 newline 0 newline 0 endbmatrix + _r^cT^-1 times beginbmatrix 0 newline dotbeta newline 0 newline 0 endbmatrix + _r^w2T^-1 times beginbmatrix dotalpha newline 0 newline 0 newline 0 endbmatrix + _r^gT^-1 times beginbmatrix 0 newline 0 newline dotdelta newline 0 endbmatrix = beginbmatrix dotgamma + dotalpha cos(beta) - dotdelta cos(alpha) sin(beta) newline omega_r2 newline omega_r3 newline 0 endbmatrix","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"omega_r2 = dotbeta cos(gamma) + dotalpha sin(beta) sin(gamma) + dotdelta sin(alpha) cos(gamma) + dotdelta cos(alpha) cos(beta) sin(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"omega_r3 = -dotbeta sin(gamma) + dotalpha sin(beta) cos(gamma) - dotdelta sin(alpha) sin(gamma) + dotdelta cos(alpha) cos(beta) cos(gamma)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_w = frac12 m_w V_w^T V_w + frac12 Omega_w^T I_w Omega_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_w = m_w g P_w(3)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_c = frac12 m_c V_c^T V_c + frac12 Omega_c^T I_c Omega_c","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_c = m_c g P_c(3)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_r = frac12 m_r V_r^T V_r + frac12 Omega_r^T I_r Omega_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_r = m_r g P_r(3)","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"T_total = T_w + T_c + T_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"P_total = P_w + P_c + P_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m = 7 n = 2","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotx) - fracpartial Lpartial x = lambda_1","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial doty) - fracpartial Lpartial y = lambda_2","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dottheta) - fracpartial Lpartial theta = tau_w - r_w cos(delta) lambda_1 - r_w sin(delta) lambda_2","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotbeta) - fracpartial Lpartial beta = -tau_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotalpha) - fracpartial Lpartial alpha = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotgamma) - fracpartial Lpartial gamma = tau_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracddt(fracpartial Lpartial dotdelta) - fracpartial Lpartial delta = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Wheel dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_11 ddotbeta + m_12 ddotgamma + m_13 ddotdelta + m_14 ddottheta + c_11 dotbeta^2 + c_12 dotgamma^2 + c_13 dotdelta^2 + c_14 dotalpha dotdelta + c_15 dotbeta dotgamma + c_16 dotbeta dotdelta + c_17 dotgamma dotdelta = tau_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Chassis longitudinal dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_21 ddotalpha + m_22 ddotbeta + m_23 ddotdelta + m_24 ddottheta + c_21 dotalpha^2 + c_22 dotdelta^2 + c_23 dotalpha dotgamma + c_24 dotalpha dotdelta + c_25 dotbeta dotgamma + c_26 dotgamma dotdelta + c_27 dotdelta dottheta + g_21 = -tau_w","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Chassis lateral dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_31 ddotalpha + m_32 ddotbeta + m_33 ddotgamma + m_34 ddotdelta + c_31 dotbeta^2 + c_32 dotgamma^2 + c_33 dotdelta^2 + c_34 dotalpha dotbeta + c_35 dotalpha dotgamma + c_36 dotbeta dotgamma + c_37 dotbeta dotdelta + c_38 dotgamma dotdelta + c_39 dotdelta dottheta = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Reaction wheel dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_41 ddotalpha + m_42 ddotgamma + m_43 ddotdelta + m_44 ddottheta + c_41 dotalpha^2 + c_42 dotbeta^2 + c_43 dotdelta^2 + c_44 dotalpha dotbeta + c_45 dotalpha dotdelta + c_46 dotbeta dotdelta + c_47 dotdelta dottheta + g_41 = tau_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"Turning dynamics:","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"m_51 ddotalpha + m_52 ddotbeta + m_53 ddotgamma + m_54 ddotdelta + m_55 ddottheta + c_51 dotalpha^2 + c_52 dotbeta^2 + c_53 dotgamma^2 + c_54 dotalpha dotbeta + c_55 dotalpha dotgamma + c_56 dotalpha dotdelta + c_57 dotalpha dottheta + c_58 dotbeta dotgamma + c_59 dotbeta dotdelta + c_510 dotgamma dotdelta + c_511 dotdelta dottheta = 0","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"fracmathrmd xleft( t right)mathrmdt = r_w cosleft( deltaleft( t right) right) fracmathrmd thetaleft( t right)mathrmdt newline fracmathrmd yleft( t right)mathrmdt = r_w sinleft( deltaleft( t right) right) fracmathrmd thetaleft( t right)mathrmdt newline fracmathrmd zleft( t right)mathrmdt = 0 newline I_w = left beginarraycccc I_w1 0 0 0 newline 0 I_w2 0 0 newline 0 0 I_w3 0 newline 0 0 0 0 newline endarray right newline I_c = left beginarraycccc I_c1 0 0 0 newline 0 I_c2 0 0 newline 0 0 I_c3 0 newline 0 0 0 0 newline endarray right newline I_r = left beginarraycccc I_r1 0 0 0 newline 0 I_r2 0 0 newline 0 0 I_r3 0 newline 0 0 0 0 newline endarray right newline mathrmw2cpTleft( t right) = left beginarraycccc 1 0 0 0 newline 0 cosleft( alphaleft( t right) right) - sinleft( alphaleft( t right) right) - r_w sinleft( alphaleft( t right) right) newline 0 sinleft( alphaleft( t right) right) cosleft( alphaleft( t right) right) r_w cosleft( alphaleft( t right) right) newline 0 0 0 1 newline endarray right newline mathrmcpgTleft( t right) = left beginarraycccc cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) 0 xleft( t right) newline sinleft( deltaleft( t right) right) cosleft( deltaleft( t right) right) 0 yleft( t right) newline 0 0 1 0 newline 0 0 0 1 newline endarray right newline mathrmw2gTleft( t right) = mathrmcpgTleft( t right) mathrmw2cpTleft( t right) newline w2P_w = left beginarrayc 0 newline 0 newline 0 newline 1 newline endarray right newline mathrmgP_wleft( t right) = mathrmw2gTleft( t right) w2P_w newline mathrmcw2Tleft( t right) = left beginarraycccc cosleft( betaleft( t right) right) 0 sinleft( betaleft( t right) right) l_c sinleft( betaleft( t right) right) newline 0 1 0 0 newline -sinleft( betaleft( t right) right) 0 cosleft( betaleft( t right) right) l_c cosleft( betaleft( t right) right) newline 0 0 0 1 newline endarray right newline mathrmcgTleft( t right) = mathrmw2gTleft( t right) mathrmcw2Tleft( t right) newline cP_c = left beginarrayc 0 newline 0 newline 0 newline 1 newline endarray right newline mathrmgP_cleft( t right) = mathrmcgTleft( t right) cP_c newline mathrmrcTleft( t right) = left beginarraycccc 1 0 0 0 newline 0 cosleft( gammaleft( t right) right) - sinleft( gammaleft( t right) right) 0 newline 0 sinleft( gammaleft( t right) right) cosleft( gammaleft( t right) right) l_cr newline 0 0 0 1 newline endarray right newline mathrmrgTleft( t right) = mathrmcgTleft( t right) mathrmrcTleft( t right) newline rP_r = left beginarrayc 0 newline 0 newline 0 newline 1 newline endarray right newline mathrmgP_rleft( t right) = mathrmrgTleft( t right) rP_r newline mathrmrw2Tleft( t right) = mathrminvleft( mathrmw2gTleft( t right) right) mathrmrgTleft( t right) newline V_wleft( t right) = mathrmbroadcastleft( D mathrmgP_wleft( t right) right) newline V_cleft( t right) = mathrmbroadcastleft( D mathrmgP_cleft( t right) right) newline V_rleft( t right) = mathrmbroadcastleft( D mathrmgP_rleft( t right) right) newline Omega_wleft( t right) = mathrmbroadcastleft( + left beginarrayc _derivativeleft( alphaleft( t right) t 1 right) newline _derivativeleft( thetaleft( t right) t 1 right) newline 0 newline 0 newline endarray right mathrminvleft( mathrmw2gTleft( t right) right) left beginarrayc 0 newline 0 newline _derivativeleft( deltaleft( t right) t 1 right) newline 0 newline endarray right right) newline Omega_cleft( t right) = mathrmbroadcastleft( + mathrmbroadcastleft( + left beginarrayc 0 newline _derivativeleft( betaleft( t right) t 1 right) newline 0 newline 0 newline endarray right mathrminvleft( mathrmcw2Tleft( t right) right) left beginarrayc _derivativeleft( alphaleft( t right) t 1 right) newline 0 newline 0 newline 0 newline endarray right right) mathrminvleft( mathrmcgTleft( t right) right) left beginarrayc 0 newline 0 newline _derivativeleft( deltaleft( t right) t 1 right) newline 0 newline endarray right right) newline Omega_rleft( t right) = mathrmbroadcastleft( + mathrmbroadcastleft( + mathrmbroadcastleft( + left beginarrayc _derivativeleft( gammaleft( t right) t 1 right) newline 0 newline 0 newline 0 newline endarray right mathrminvleft( mathrmrcTleft( t right) right) left beginarrayc 0 newline _derivativeleft( betaleft( t right) t 1 right) newline 0 newline 0 newline endarray right right) mathrminvleft( mathrmrw2Tleft( t right) right) left beginarrayc _derivativeleft( alphaleft( t right) t 1 right) newline 0 newline 0 newline 0 newline endarray right right) mathrminvleft( mathrmrgTleft( t right) right) left beginarrayc 0 newline 0 newline _derivativeleft( deltaleft( t right) t 1 right) newline 0 newline endarray right right) newline T_wleft( t right) = mathrmadjointleft( V_wleft( t right) right) mathrmbroadcastleft( * V_wleft( t right) mathrmRefleft( 05 m_w right) right)_1 + mathrmadjointleft( Omega_wleft( t right) right) mathrmbroadcastleft( * I_w Omega_wleft( t right) 05 right)_1 newline P_wleft( t right) = g mathrmgP_wleft( t right)_3 m_w newline T_cleft( t right) = mathrmadjointleft( V_cleft( t right) right) mathrmbroadcastleft( * V_cleft( t right) mathrmRefleft( 05 m_c right) right)_1 + mathrmadjointleft( Omega_cleft( t right) right) mathrmbroadcastleft( * I_c Omega_cleft( t right) 05 right)_1 newline P_cleft( t right) = g mathrmgP_cleft( t right)_3 m_c newline T_rleft( t right) = mathrmadjointleft( V_rleft( t right) right) mathrmbroadcastleft( * V_rleft( t right) mathrmRefleft( 05 m_r right) right)_1 + mathrmadjointleft( Omega_rleft( t right) right) mathrmbroadcastleft( * I_r Omega_rleft( t right) 05 right)_1 newline P_rleft( t right) = g mathrmgP_rleft( t right)_3 m_r newline T_totalleft( t right) = T_rleft( t right) + T_cleft( t right) + T_wleft( t right) newline P_totalleft( t right) = P_wleft( t right) + P_cleft( t right) + P_rleft( t right) newline Lleft( t right) = T_totalleft( t right) - P_totalleft( t right) newline","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"L = 05 left( left( fracfracmathrmd alphaleft( t right)mathrmdt cosleft( betaleft( t right) right)sin^2left( betaleft( t right) right) + cos^2left( betaleft( t right) right) + fracleft( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + left( - sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) fracmathrmd deltaleft( t right)mathrmdtsinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( betaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) right)^2 I_c1 + left( fracmathrmd betaleft( t right)mathrmdt + fracleft( - left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - left( - sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtsinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( betaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) right)^2 I_c2 + left( fracsinleft( betaleft( t right) right) fracmathrmd alphaleft( t right)mathrmdtsin^2left( betaleft( t right) right) + cos^2left( betaleft( t right) right) + fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) fracmathrmd deltaleft( t right)mathrmdtsinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) + cosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( betaleft( t right) right) right) + left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) + cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right) right) right)^2 I_c3 right) + 05 left( left( fracmathrmd gammaleft( t right)mathrmdt + fracleft( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) fracmathrmd alphaleft( t right)mathrmdtleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( frac - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + fracleft( left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) sinleft( betaleft( t right) right) right) right)^2 I_r1 + left( fracleft( - left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) - left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) fracmathrmd alphaleft( t right)mathrmdtleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( frac - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + fracleft( - left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) sinleft( betaleft( t right) right) right) + fracfracmathrmd betaleft( t right)mathrmdt cosleft( gammaleft( t right) right)sin^2left( gammaleft( t right) right) + cos^2left( gammaleft( t right) right) right)^2 I_r2 + left( fracleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) fracmathrmd alphaleft( t right)mathrmdtleft( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( frac - left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( frac - left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( left( frac - sinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fraccosleft( alphaleft( t right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + left( fracsinleft( deltaleft( t right) right) left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) cosleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) left( fracsinleft( deltaleft( t right) right) left( - sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right) left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + frac - left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) right) left( fracleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracleft( cos^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sinleft( deltaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right)cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right) + fracleft( left( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) fracmathrmd deltaleft( t right)mathrmdtleft( cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) - left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) sinleft( gammaleft( t right) right) right) left( - cosleft( alphaleft( t right) right) sinleft( betaleft( t right) right) left( cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - cosleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( - sinleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) + left( sinleft( deltaleft( t right) right) cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) + left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) cosleft( gammaleft( t right) right) right) left( left( cosleft( gammaleft( t right) right) sinleft( alphaleft( t right) right) + cosleft( alphaleft( t right) right) sinleft( gammaleft( t right) right) cosleft( betaleft( t right) right) right) left( sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) + sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) - cosleft( alphaleft( t right) right) left( - cosleft( alphaleft( t right) right) cosleft( gammaleft( t right) right) cosleft( deltaleft( t right) right) - sinleft( gammaleft( t right) right) left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) sinleft( betaleft( t right) right) right) + frac - fracmathrmd betaleft( t right)mathrmdt sinleft( gammaleft( t right) right)sin^2left( gammaleft( t right) right) + cos^2left( gammaleft( t right) right) right)^2 I_r3 right) + 05 left( fracleft( fracmathrmd deltaleft( t right)mathrmdt right)^2 left( sin^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) cosleft( alphaleft( t right) right) right)^2 I_w3left( cos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) right)^2 + left( fracmathrmd alphaleft( t right)mathrmdt right)^2 I_w1 + left( fracleft( sin^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) + cos^2left( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) fracmathrmd deltaleft( t right)mathrmdtcos^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) sin^2left( deltaleft( t right) right) + left( cos^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) + sin^2left( alphaleft( t right) right) cosleft( deltaleft( t right) right) right) cosleft( deltaleft( t right) right) + fracmathrmd thetaleft( t right)mathrmdt right)^2 I_w2 right) + 05 m_c left( left( fracmathrmdmathrmdt left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt left( xleft( t right) + l_c sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + r_w sinleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) + l_c sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt 1 right)^2 + left( fracmathrmdmathrmdt left( yleft( t right) + l_c sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - r_w cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) - l_c cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 right) + 05 m_r left( left( fracmathrmdmathrmdt left( xleft( t right) + l_c sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + r_w sinleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) + l_c sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) + l_cr left( sinleft( betaleft( t right) right) cosleft( deltaleft( t right) right) + sinleft( deltaleft( t right) right) cosleft( betaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right)^2 + left( fracmathrmdmathrmdt left( yleft( t right) + l_c sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - r_w cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) - l_c cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) + l_cr left( sinleft( deltaleft( t right) right) sinleft( betaleft( t right) right) - cosleft( betaleft( t right) right) cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right) right)^2 + left( fracmathrmdmathrmdt 1 right)^2 + left( fracmathrmdmathrmdt left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) + l_cr cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) right)^2 right) + 05 m_w left( left( fracmathrmdmathrmdt r_w cosleft( alphaleft( t right) right) right)^2 + left( fracmathrmdmathrmdt left( yleft( t right) - r_w cosleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt left( xleft( t right) + r_w sinleft( deltaleft( t right) right) sinleft( alphaleft( t right) right) right) right)^2 + left( fracmathrmdmathrmdt 1 right)^2 right) - g m_w r_w cosleft( alphaleft( t right) right) - g left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) m_c - g left( r_w cosleft( alphaleft( t right) right) + l_c cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) + l_cr cosleft( alphaleft( t right) right) cosleft( betaleft( t right) right) right) m_r","category":"page"},{"location":"reactionwheelunicycle.html","page":"Reaction Wheel Unicycle","title":"Reaction Wheel Unicycle","text":"left beginarrayc _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline _derivativeleft( 0 t 1 right) newline endarray right = left beginarrayc lambda_1 newline lambda_2 newline tau_w - r_w sinleft( deltaleft( t right) right) lambda_2 - r_w cosleft( deltaleft( t right) right) lambda_1 newline -tau_w newline 0 newline tau_p newline 0 newline endarray right","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"Description = \"Read the documentation of the Porta.jl project.\"","category":"page"},{"location":"index.html#Geometrize-the-quantum!","page":"Home","title":"Geometrize the quantum!","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"This project is inspired by Eric Weinstein's Graph-Wall-Tome (GWT) project. Watch visual models on the YouTube channel.","category":"page"},{"location":"index.html#Requirements","page":"Home","title":"Requirements","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"CSV v0.10.13\nDataFrames v1.6.1\nFileIO v1.16.3\nGLMakie v0.9.9","category":"page"},{"location":"index.html#Installation","page":"Home","title":"Installation","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"You can install Porta by running this (in the REPL):","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"]add Porta","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"or,","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"Pkg.add(\"Porta\")","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"or get the latest experimental code.","category":"page"},{"location":"index.html","page":"Home","title":"Home","text":"]add https://github.com/iamazadi/Porta.jl.git","category":"page"},{"location":"index.html#Usage","page":"Home","title":"Usage","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"For client-side code read the tests, and for examples on how to build, please check out the models directory. See planethopf.jl as an example.","category":"page"},{"location":"index.html#Status","page":"Home","title":"Status","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"Logic [Doing]\nSet Theory [TODO]\nTopology [TODO]\nTopological Manifolds [TODO]\nDifferentiable Manifolds [TODO]\nBundles [TODO]\nGeometry: Symplectic, Metric [TODO]\nDocumentation [TODO]\nGeometric Unity [TODO]","category":"page"},{"location":"index.html#References","page":"Home","title":"References","text":"","category":"section"},{"location":"index.html","page":"Home","title":"Home","text":"Physics and Geometry, Edward Witten, (1987)\nThe iconic Wall of Stony Brook University\nThe Road to Reality, Sir Roger Penrose, (2004)\nA Portal Special Presentation- Geometric Unity: A First Look\nPlanet Hopf, Dror Bar-Natan, (2010)\nSPINORS AND SPACE-TIME, Volume 1: Two-spinor calculus and relativistic fields, Roger Penrose, Wolfgang Rindler, (1984)\nA Young Person's Guide to the Hopf Fibration, Zachary Treisman, (2009)\nMathematical Gauge Theory, with Applications to the Standard Model of Particle Physics, Mark J.D. Hamilton, (2018)\nDynamics in the Hopf bundle, the geometric phase and implications for dynamical systems, Rupert Way, (2008)","category":"page"}] }