Separation of Variable, spherical coordinates.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Separation of Variables in spherical coordinates\n",
    "\n",
    "with the Laplace equation in spherical coordiantes:\n",
    "\n",
    "$$ \\frac{1}{r^2}\\frac{\\partial}{\\partial r}\\left (r^2 \\frac{\\partial V}{\\partial r}\\right )+ \n",
    "   \\frac{1}{r^2\\sin \\theta}\\frac{\\partial}{\\partial \\theta} \\left (\\sin \\theta \\frac{\\partial V}{\\partial \\theta}\\right )+\n",
    "   \\frac{1}{r^2\\sin^2 \\theta}\\frac{\\partial^2 V}{\\partial \\phi^2}\n",
    "   =0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "we are assuming azimuthal symmetry here and ths V is independent of $\\phi$, even though the general care is solveable as well and it simplifies to:\n",
    "\n",
    "$$ \\frac{\\partial}{\\partial r}\\left (r^2 \\frac{\\partial V}{\\partial r}\\right )+ \n",
    "   \\frac{1}{\\sin \\theta}\\frac{\\partial}{\\partial \\theta} \\left (\\sin \\theta \\frac{\\partial V}{\\partial \\theta}\\right )\n",
    "   =0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again we are looking for solutions as products:\n",
    "    \n",
    "$$ V(r,\\theta) = R(r)\\Theta(\\theta)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Separating of variables gives us:\n",
    "    \n",
    "$$ \\frac 1 R \\frac{\\partial}{\\partial r}\\left (r^2 \\frac{\\partial R}{\\partial r}\\right )= \n",
    "   -\\frac{1}{\\Theta\\sin \\theta}\\frac{\\partial}{\\partial \\theta} \\left (\\sin \\theta \\frac{\\partial \\Theta}{\\partial \\theta}\\right ) $$\n",
    "   \n",
    "which can only be equal if the left side and the right side are constants (since they each only depend on $r$ or $\\theta$ respectively)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we are setting constants for the left side:\n",
    "\n",
    "    \n",
    "$$ \\frac 1 R \\frac{\\partial}{\\partial r}\\left (r^2 \\frac{\\partial R}{\\partial r}\\right )=\n",
    "   l(l+1)$$\n",
    "   \n",
    "and for the right side:\n",
    "$$ \\frac{1}{\\Theta\\sin \\theta}\\frac{\\partial}{\\partial \\theta} \\left (\\sin \\theta \\frac{\\partial \\Theta}{\\partial \\theta}\\right ) = - l(l+1) $$\n",
    "\n",
    "and thus have converted a PDE in two ODEs."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving the left side:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.interactive import printing # use latex for printing\n",
    "printing.init_printing(use_latex=True)\n",
    "from sympy import Function, dsolve, Eq, Derivative, sin, cos, symbols # import necessary methods\n",
    "from sympy.abc import theta # import variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- l \\left(l + 1\\right) R{\\left (r \\right )} + \\frac{d}{d r} r^{2} \\frac{d}{d r} R{\\left (r \\right )} = 0$$"
      ],
      "text/plain": [
       "                  d ⎛ 2 d       ⎞    \n",
       "-l⋅(l + 1)⋅R(r) + ──⎜r ⋅──(R(r))⎟ = 0\n",
       "                  dr⎝   dr      ⎠    "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r, l = symbols('r l',positive=True)\n",
    "R = Function('R',real=True)\n",
    "s1_ode = Eq(Derivative((r**2*Derivative(R(r),r,1)), r, 1)-l*(l+1)*R(r))\n",
    "s1_ode"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$R{\\left (r \\right )} = C_{1} r^{l} + C_{2} r^{- l - 1}$$"
      ],
      "text/plain": [
       "           l       -l - 1\n",
       "R(r) = C₁⋅r  + C₂⋅r      "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sol1 = dsolve(s1_ode, R(r))\n",
    "sol1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Solving the right side"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$l \\left(l + 1\\right) \\Theta{\\left (\\theta \\right )} \\sin{\\left (\\theta \\right )} + \\frac{d}{d \\theta} \\sin{\\left (\\theta \\right )} \\frac{d}{d \\theta} \\Theta{\\left (\\theta \\right )} = 0$$"
      ],
      "text/plain": [
       "                        d ⎛       d       ⎞    \n",
       "l⋅(l + 1)⋅Θ(θ)⋅sin(θ) + ──⎜sin(θ)⋅──(Θ(θ))⎟ = 0\n",
       "                        dθ⎝       dθ      ⎠    "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t, l = symbols('theta l')\n",
    "T = Function('Theta',real=True)\n",
    "s2_ode = Eq(Derivative(sin(t)*Derivative(T(t),t,1), t, 1)+l*(l+1)*sin(t)*T(t))\n",
    "s2_ode"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "ename": "NotImplementedError",
     "evalue": "solve: Cannot solve l*(l + 1)*Theta(theta)*sin(theta) + sin(theta)*Derivative(Theta(theta), (theta, 2)) + cos(theta)*Derivative(Theta(theta), theta)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNotImplementedError\u001b[0m                       Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-5-a864d814468b>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0msol2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdsolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ms2_ode\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      2\u001b[0m \u001b[0msol2\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.7/site-packages/sympy/solvers/ode.py\u001b[0m in \u001b[0;36mdsolve\u001b[0;34m(eq, func, hint, simplify, ics, xi, eta, x0, n, **kwargs)\u001b[0m\n\u001b[1;32m    628\u001b[0m         hints = _desolve(eq, func=func,\n\u001b[1;32m    629\u001b[0m             \u001b[0mhint\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msimplify\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mxi\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mxi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0meta\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0meta\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtype\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m'ode'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mics\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mics\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 630\u001b[0;31m             x0=x0, n=n, **kwargs)\n\u001b[0m\u001b[1;32m    631\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    632\u001b[0m         \u001b[0meq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpop\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'eq'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0meq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m/usr/local/lib/python3.7/site-packages/sympy/solvers/deutils.py\u001b[0m in \u001b[0;36m_desolve\u001b[0;34m(eq, func, hint, ics, simplify, **kwargs)\u001b[0m\n\u001b[1;32m    233\u001b[0m             \u001b[0;32mraise\u001b[0m \u001b[0mValueError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstring\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0meq\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\" does not match hint \"\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    234\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 235\u001b[0;31m             \u001b[0;32mraise\u001b[0m \u001b[0mNotImplementedError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdummy\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\"solve\"\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\": Cannot solve \"\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mstr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0meq\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    236\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mhint\u001b[0m \u001b[0;34m==\u001b[0m \u001b[0;34m'default'\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    237\u001b[0m         return _desolve(eq, func, ics=ics, hint=hints['default'], simplify=simplify,\n",
      "\u001b[0;31mNotImplementedError\u001b[0m: solve: Cannot solve l*(l + 1)*Theta(theta)*sin(theta) + sin(theta)*Derivative(Theta(theta), (theta, 2)) + cos(theta)*Derivative(Theta(theta), theta)"
     ]
    }
   ],
   "source": [
    "sol2 = dsolve(s2_ode, T(t))\n",
    "sol2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Legendre Polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.plotting import plot\n",
    "from sympy import legendre, pi\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = symbols('theta',positive=True)\n",
    "P0 = lambda t: legendre(0,t)\n",
    "P1 = lambda t: legendre(1,t)\n",
    "P2 = lambda t: legendre(2,t)\n",
    "P3 = lambda t: legendre(3,t)\n",
    "P4 = lambda t: legendre(4,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOBAMAAADgeEClAAAAIVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADdcGRXAAAACnRSTlMAMt0Qq5nNdrvvxbMB0AAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABtJREFUCB1jYGBUZmAwCVvMwMDAThbB0rnIHQBAqQoWQ8kakgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P0(cos(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\cos{\\left (\\theta \\right )}$$"
      ],
      "text/plain": [
       "cos(θ)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P1(cos(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{3 \\cos^{2}{\\left (\\theta \\right )}}{2} - \\frac{1}{2}$$"
      ],
      "text/plain": [
       "     2       \n",
       "3⋅cos (θ)   1\n",
       "───────── - ─\n",
       "    2       2"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P2(cos(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{5 \\cos^{3}{\\left (\\theta \\right )}}{2} - \\frac{3 \\cos{\\left (\\theta \\right )}}{2}$$"
      ],
      "text/plain": [
       "     3              \n",
       "5⋅cos (θ)   3⋅cos(θ)\n",
       "───────── - ────────\n",
       "    2          2    "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P3(cos(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{35 \\cos^{4}{\\left (\\theta \\right )}}{8} - \\frac{15 \\cos^{2}{\\left (\\theta \\right )}}{4} + \\frac{3}{8}$$"
      ],
      "text/plain": [
       "      4            2       \n",
       "35⋅cos (θ)   15⋅cos (θ)   3\n",
       "────────── - ────────── + ─\n",
       "    8            4        8"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P4(cos(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x10e3501d0>"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(P0(cos(t)),P1(cos(t)),P2(cos(t)),P3(cos(t)),P4(cos(t)),(t,0,pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Legendre polynomials with the argument $\\cos\\theta$ are the solutions for the right side:\n",
    "\n",
    "$$ \\Theta(\\theta) = P_l(\\cos\\theta) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which gives the most general separabe solution to Laplace equation (for physics):\n",
    "\n",
    "$$ V(r,\\theta)=\\sum_{l=0}^{\\infty}\\left (A_l r^l+\\frac{B_l}{r^{l+1}}\\right ) P_l(\\cos\\theta) $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}