{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Allgemeine Relativitätstheorie mit dem Computer \n",
    "# General Theory of Relativity on the Computer\n",
    "\n",
    "## Vorlesung gehalten an der J.W.Goethe-Universität in Frankfurt am Main (Wintersemester 2026/27)\n",
    "\n",
    "### von Dr.phil.nat. Dr.rer.pol. Matthias Hanauske\n",
    "\n",
    "### Frankfurt am Main 23.09.2026\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bewegung eines Probekörpers um ein nicht-rotierendes schwarzes Loch in der Ebene \n",
    "## Teil II: Klassifizierung unterschiedlicher Bahnbewegungen"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Im Folgenden wird die <i>Geodätengleichung</i> in vorgegebener Schwarzschild Raumzeit betrachtet. Die Geodätengleichung beschreibt wie sich ein Probekörper (Masse Probekörper $m << M$ Masse schwarzes Loch) im Raum bewegt und sagt voraus, dass diese Bewegung sich stets entlang der extremalsten Kurve, in der durch die Metrik beschriebenen gekrümmten Raumzeit, vollzieht. \n",
    "\n",
    "Die Geodätengleichung lässt sich durch folgendes Variationsprinzip herleiten\n",
    "\n",
    "\\begin{equation}\n",
    "\\int_{x^\\mu(A)}^{x^\\mu(B)} ds= \\int_{x^\\mu(A)}^{x^\\mu(B)} \\sqrt{g_{\\mu\\nu} dx^\\mu dx^\\nu}=\\int_A^B \\sqrt{g_{\\mu\\nu} \\frac{dx^\\mu}{d\\lambda} \\frac{dx^\\nu}{d\\lambda}} d\\lambda \\rightarrow \\hbox{Extremal}\n",
    "\\end{equation}\n",
    "\n",
    ", wobei man, bei festgehaltenen Anfangs- und Endpunkten ($x^\\mu(A)$ und $x^\\mu(B)$), nach den extremalsten Kurven $x^\\mu(\\lambda)$ in der Raumzeit sucht. Den Parameter $\\lambda$ bezeichnet man als affiner Parameter, welcher z.B. die Eigenzeit $\\tau$ des Körpers sein kann. Die Geod&auml;tengleichung ergibt sich dann mittels der Euler-Lagrange Gleichungen, wobei man als Lagrangedichte den Ausdruck $L = \\sqrt{g_{\\mu\\nu} \\frac{dx^\\mu}{d\\lambda} \\frac{dx^\\nu}{d\\lambda}}$, bzw. alternativ $L = g_{\\mu\\nu} \\frac{dx^\\mu}{d\\lambda} \\frac{dx^\\nu}{d\\lambda}$ verwendet.\n",
    "\n",
    "\\begin{equation}\n",
    "\\frac{d}{d\\lambda} \\left( \\frac{\\partial L}{\\partial \\frac{d x^\\alpha}{d \\lambda}} \\right) -\n",
    "    \\frac{\\partial L}{\\partial x^\\alpha} = 0 \\quad \\xrightarrow{\\cdot \\frac{1}{2}g^{\\mu \\alpha }} \\quad \\frac{d^2 x^\\mu}{d\\lambda^2} +\n",
    "    \\Gamma^\\mu_{\\nu \\rho} \\frac{d x^\\nu}{d\\lambda} \\frac{d x^\\rho}{d\\lambda} ~=~ 0\n",
    "\\end{equation}\n",
    "\n",
    "$\\Gamma^\\mu_{\\nu \\rho}$ sind die Christoffel Symbole zweiter Art und $\\lambda$ ein affiner Parameter \n",
    "(z.B. die Eigenzeit $\\tau$ des Probekörpers). Die Geodätengleichung ist ein System gekoppelter Differenzialgleichungen, das wir uns im Folgenden näher betrachten werden."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir betrachten im Folgenden die Geod&auml;tengleichung in vorgegebener Schwarzschild-Raumzeit\n",
    "\n",
    "$$ g_{\\mu\\nu}=\\left( \\begin{array}{ccc} 1-\\frac{2\\,M}{r} & 0 & 0 & 0\\\\ 0&  -\\frac{1}{1-\\frac{2\\,M}{r}}&  0&0 \\\\ 0&  0&  -r^2& 0\\\\ 0&  0&  0& -r^2 \\hbox{sin}^2(\\theta)\\\\ \\end{array} \\right) \n",
    "$$\n",
    "\n",
    ", wobei wir wieder ein sph&auml;risches Koordinatensystem benutzen ($x^\\mu=\\left(t,r,\\theta,\\phi \\right)$). Die Geod&auml;tengleichung stellt ein System gekoppelter nichtlinearer Differenzialgleichungen dar\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "&& \\frac{d^2 t}{d\\lambda^2} = -  \\Gamma^0_{\\nu \\rho} \\frac{d x^\\nu}{d\\lambda} \\frac{d x^\\rho}{d\\lambda} \\\\\n",
    "&& \\frac{d^2 r}{d\\lambda^2} = -  \\Gamma^1_{\\nu \\rho} \\frac{d x^\\nu}{d\\lambda} \\frac{d x^\\rho}{d\\lambda}\\\\\n",
    "&& \\frac{d^2 \\theta}{d\\lambda^2} = -  \\Gamma^2_{\\nu \\rho} \\frac{d x^\\nu}{d\\lambda} \\frac{d x^\\rho}{d\\lambda}\\\\\n",
    "&& \\frac{d^2 \\phi}{d\\lambda^2} = -  \\Gamma^3_{\\nu \\rho} \\frac{d x^\\nu}{d\\lambda} \\frac{d x^\\rho}{d\\lambda} \\quad ,\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "wobei $\\lambda$ ein affiner Parameter (z.B. die Eigenzeit $\\tau$), t, r, $\\theta$ und $\\phi$ die Koordinaten und $\\Gamma^\\mu_{\\nu \\rho}$ die Christoffel Symbole zweiter Art darstellen."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Zunächst wird das Python-Modul \"EinsteinPy\" eingebunden, welches auf dem Modul SymPy basiert und symbolische Berechnungen in der Allgemeinen Relativitätstheorie relativ einfach möglich macht. Dann definieren wir die  zugrundeliegenden Koordinaten und die kovariante Form der Raumzeit-Metrik der Schwarzschildmetrik."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "from einsteinpy.symbolic import *"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "t, r, theta, phi, M = symbols('t, r, theta, phi, M')\n",
    "Metric = diag((1-2*M/r), -1/(1-2*M/r), -r**2, -r**2*sin(theta)**2).tolist()\n",
    "g = MetricTensor(Metric, [t, r, theta, phi])\n",
    "chr = ChristoffelSymbols.from_metric(g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Geodätische Bewegung eines Probekörpers\n",
    "\n",
    "Um die Geodätengleichung \n",
    "\n",
    "$$\n",
    "\\frac{d^2 x^\\mu}{d\\tau^2} +  \\Gamma^\\mu_{\\nu \\rho} \\frac{d x^\\nu}{d\\tau} \\frac{d x^\\rho}{d\\tau} ~=~ 0 \n",
    "$$\n",
    "\n",
    "zu berechnen, definieren zunächst den ersten Term der Gleichung $\\frac{d^2 x^\\mu}{d\\tau^2}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "tau = symbols('tau')\n",
    "x0 = Function('t')(tau)\n",
    "x1 = Function('r')(tau)\n",
    "x2 = Function('theta')(tau)\n",
    "x3 = Function('phi')(tau)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "xmu = GenericVector([x0, x1, x2, x3],[t, r, theta, phi], config='u',parent_metric=Metric)\n",
    "Geod1 = diff(xmu.tensor(),tau,tau)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "und nun den zweiten Term $\\Gamma^\\mu_{\\nu \\rho} \\frac{d x^\\nu}{d\\tau} \\frac{d x^\\rho}{d\\tau}$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "Geod2T = tensorproduct(chr.tensor(),diff(xmu.tensor(),tau),diff(xmu.tensor(),tau))\n",
    "Geod2 = tensorcontraction(Geod2T, (1, 3),(2,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Geodätengleichung ist ein System von nichtlinearen gekoppelten Differenzialgleichungen zweiter Ordnung. Wir definieren die entsprechenden vier Gleichungen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "Geod = Geod1 + Geod2\n",
    "GeodEq0 = Eq(Geod[0],0)\n",
    "GeodEq1 = Eq(Geod[1],0)\n",
    "GeodEq2 = Eq(Geod[2],0)\n",
    "GeodEq3 = Eq(Geod[3],0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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21W1pvwRcTNDptQSB0UEgzi1UbK5h19kD2x0m+ywgulBEM5J7faqcMq5eAadcPu82R3/1jPdmk5eJTF6XQaPaVFAjf2QOrycHGgs8HsYQg0FgkpoehsVC+bYNYrFia0UZcQiqExsg6nPRwWKLHqHz3LZuMFACo3pCGgwU2YzIldIcjtlhLquLTdYSmdSZcyA5kH4rGkurR+yy2wqHwplBcium2t94huL6Pmu+6AD5I/OTM0KSB+PDGGN8bPwYENLckLlxU8yYDvlSHaMGbluglHSLjy227nn7008CoxohdcnAvPFdM7BhUDTQIe3LcxGGyylGWqj24P6vAnFf4Io865Uon5WMg8nSNo0+2Ovb6ofSkhdP1mfKcwZMMfnQ1vs8SG2gAcOVAnLYqTwnM+go0Uz/NfJnUg6RK+QdA+DxcOmQLn7I3F67QM7oSNATPaO7NwpZUPl32YIts0oCoxohP5g7xTaxbEJUERc3Ek6UYowbCOhj33+l2G2f4n460tDgPDu1ZRsWFFlp4zm4+B24oCNnZMgH9+AX9dTOTSTFRktf+ReNfxcPKy/nsL0xHpINxpYFo22x4EyzrVzFG4wlgYuxECV4WKlrrj+/qB1KMhn4CX5fcfAmlN57P0N5rIYNd1vPeAzUdTQqjl18DBpvH4cVVc9LwVD5ww+yPxnwYwRfqTFxt4AqbxinhHEWgbFfRky62B5NAlMZIVbi2Euw/rpiN8m7KtWWS/EwLI8Uu4PzCA9Kl05CeEgVFfoeKjwTjpS/XH1VXQSGyh9PKDaoixA9UacpXywWbqHQGHKriVcUwD+z0KxhMQl0nXLicmzmbBCFN12B+nTFwD/qU3FoHU8Xh5S8iWs3IYbmsfLc2Y5lKOY6Ns17r3wMVsMAqR7GLbfqKnteEC1MKsIQ+bPNPalJJznglbKIBCOjZ8aOsbLFhVdIGmOpMuqkhktZG0wlgdGNkAhl0NmK2UAPof21KmMopgAmJbTZuzxxH2ECim76R1FZDdmOcQ1rB5vUCzctysfj4FyH+gBlrLRPFd+4nPn/1Mjf+Jif2ml7ZFv6h8aCl//uKaBfjA9fdJOXLkbKcotgLp+yDSaQwOq+opdycMVfvNWaQAarQykZsBpjBFNPbHRa1QdGi+tpe5Vi9D42hJsE2iRw0Va4UBnXqtlbo4XoWaJbvKi5PCnOx5D5BpsEFpHA6jwhpKBVma3TUX3/ssjoHdip94LYbrJF2WCTwCISWKMnhCDYy2/78ulVAhmfu9c5vZS3HlolsEpPCIr9Kr39/Gjr8NUXSr4czPLJynYTVC/GreUIEvg/sXRSjuDFeM4AAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle \\frac{2 M \\frac{d}{d \\tau} r{\\left(\\tau \\right)} \\frac{d}{d \\tau} t{\\left(\\tau \\right)}}{r \\left(- 2 M + r\\right)} + \\frac{d^{2}}{d \\tau^{2}} t{\\left(\\tau \\right)} = 0$"
      ],
      "text/plain": [
       "    d        d                       \n",
       "2⋅M⋅──(r(τ))⋅──(t(τ))    2           \n",
       "    dτ       dτ         d            \n",
       "───────────────────── + ───(t(τ)) = 0\n",
       "    r⋅(-2⋅M + r)          2          \n",
       "                        dτ           "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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hPEzWCBiBGRHQRMZ5Rt6c9UrkjLialBEwAvtHYIh+9Era/vvZLTACRsAIGAEjYAQOiMC9f//73xxc42A4/lB38aV/WYRvhxZ2PiNwawhIPvhG3GGd2sdr+qnt8PjJldeJxv+lcofeSrZeTPS6o4xAAoEj68gp+tHbnYnB4igjYATmQUDKydud80BpKkbACBwMgSH60dudB+t0N8cIGAEjYASMgBE4BgIPjtEMt8IIGAEjYASMgBGICGiVhiNML8M9b97jniv+4q37uyT/bhEBG2lb7JWZeJIwIpicE+I8EOGnuvi2z66+VSae7YyAETACRqAMga+l6+u/u1OYc6P8z/KHZWSce00EvN25JvrL180HaD+UcPKfk3wC4XddvCRiZwSMgBEwAsdG4IX0fvNlHuYDPofDR8ftdoKAV9J20lEj2XzVKve+7lNv2bWy+dYIGAEjYAR2jgCraH/svA03z76NtAMPAT0xtbc1edPu4u+dDgyBm7Y+Apx/8RmY9fvBHNwYAtL/37WajNH2T2JeaGXz7RUR6NWP/gTHFXtjraoklF+pbv6q6QeF/eX3tTrC9RqBG0NA+oa/i8Mw+HXOpoveQ9HrPRSvfGztkY8HVP6+Lqn/FM95LXjFsPlW9+0HXEVvx8Gvrvq8WR9nygsOP+n6SGE/NPUBtqF0n0nbUGcsxYqEEsX0sa5PFcZgszMCRsAILIqAdA2GEUbBrAZaYJpD8RhdXHwQmWMcHIo/c0rD2PpFF/+zyYtTF055OLfFR5d/VfjLUOYi38YiaD/t6nXKx0tj7KDYQOtFa3sZbKRtr08W4UiCytMTf4aLcPvg6CIom6gR2D8CTOq6OHT+ma6vdP2kq0hnKH+10iV/8GpPIXIlh+I/Eu0fdF20IfAJr1yDjB7lW92J73/EBP3S+dCtdAw0DFke0N9wH+JWb4MZGIaAz6QNw2lTuSRkKBu2Lx8rjPBVgqh7DLG/df+dLpTO/9P1scJx6Z503BNdMa6K8I8RMAJGICDAqgtblNWZJvkYArwV/l5IH+KxtcYW4lKu9FA8+g492XZf0E5d8MtD7G5c4Dvq+6jba/6VTnvpA4y0aKCC267aWTfoRgP3b7Tde2/2MwldJWjyEUKW6OMTa/VigO4R2te6eOKKjmV94n+MEfaNgBEwAi0EXumelafoit4Kl+7BOHgiv31wPdKb7EM76LhIC/13cSg+8PKn/EoPhvuqjMLoQ7Y4KwNG/qYeXOFLF0bYw9jIhI/+/z4RTxTbv7QRP16sQF4YdEq32ygCXknbaMfk2JKAoQD/DumP5LN8HT9OiDA3FSNnNV4q/b8hP2cyfC4hgGHPCBiBSwSkL9rGSulb4TxANvXQZSUzxohfjKzq/FuCbGWIhXjaRV6MOXTlQ/mEWSlc4tycyJa5wBdGFw/Y7Hig77scOHOEhbacGV+6L1n57KrDaSsiYCNtRfBHVo1yiSthKJy4gnaSUNZhaOsepdRWuCTZGQEjYAQ6EZD+wHh5pos3CUuMLt6S/LST+EyJ4gsjht2D3MMn8ZF3vhnGgyovEVTbnPJx8DrbeTTVh2HIw3OslzoGOZXB0OLh+qQw+KPjs478utDxX+gqri9L2AmbQeD+ZjgxI4MQQCCDYKIIcNFgu7vzrxEwAkZgBgSkZ4rfCleZaFQs/hFV1YWBVnIonh0IthCbq2sg1b4nboqrVummECgsC9aVYVdYztl3gICNtB10UoZFhJKl+rMl7kxeRxsBI2AEihEI+oXty6FvhWPwLK6XgoHGeSxW+TC8MA7hk23CyoU88WgIcaw4sYVYbXMSEcrh73nHgTbSLrsDIvDggG26lSahDFm2tzMCRsAI9CIQDBK2L5d8K5ztxGs8OHIQnhUr/NqpjdWRD/lsFRKutgN1z5kzVpz+ULjSmyEPW53kwQjFyNujw8DE8OS6BvZ7xGi3PNtI22HXSRBZ5uea7RzFDmEwy0bACJQhUL0VLv3xiy5WoTBOeDOcM11c1RuTumc1auxb4RhOlF/UicfOQ/FKZ6uWq3aKw4Cpz8ql8tSZ9xWIeDMnYLDZHQgBG2n77EyEkS2FTbyRtE8IzbURuB0EpCvQGXHr75HCS70VDu2mgXc7IK/X0rh6BvZ2B0PARtoOOzQYZ/GzGztsgVk2AkbgygiwwhVfMuL8Vv0muPRJHYYn3bMaM3ZFJruSJrqk/aYLf6j7PPBT59f92/pmwYDquddFXumsRnLspO0qY0npZ7iGTLz4Nfch/7iS1ubD9wdAwEbaATrRTTACRsAIdCEgw6AyuuRHoyIabF3FxqS9UaHkio7qJo2/aJrkRKfTeJpEvKCw+EgZYSfF8802VirPtlsLSJdmjXjbWCtFbgf57++AR7NoBIyAETAC8yDgt8LnwXFLVOLKJEaw3cEQsJF2sA51c4yAETACHQiwkrbkW+GcR4tGQwcbTpoRAa+kzQjm1kjZSNtaj5gfI2AEjMACCGj7jZcHuJZ8K/x30fc3u+bpP/4zFReNsLu7y1/6lDd1vZJ2ic3uY3wmbfdd6AYYASNgBAYhwGS+9FvhnH3zN7sGdUc6k4ytn0JKPD/4k+JYoeTTKam/fuKcn9/0T8O5+1gbabvvQjfACBgBI9CPgCZ4JvJF3wqnDl2s6LCatqrhID44wB//l5PVKLZhaT8frt3sZ0LEW+nbn2DNm6Z2B0Tg/gHb5CYZASNgBIzAegjw5mj90dj12Ki2CVmV+h4eZPzwtiUGzWPuF3QYqVfZelSbMDxpz1Jv6y4Ik0kPQcAraUNQch4jYASMgBEYigCrOnwPbbW/WZLxguHCSh5bhmzxxm3C5wpXnyNR/CJO9K+5gviFGvGz6ryKUbgIYCbaiYBX0jrhcaIRMAJGwAiUIBCMILY92W5cxcGDLrY0WdH7ITIReIu3R/AxhFczho8A4NbbYCNt6z1k/oyAETAC+0PguVh+uQG2WUlbdOVsrTbK4Hyhur8NxuhabLjehRG49/btVf5hY+FmmLwRMAJGwAhsCQEZD6ykPZW/ykqP6uVt1r/lb+IfCubsm9A2DLQtnP2bs2mm1ULAK2ktQHxrBIyAETAC0xGQAcFHczGSWM1aw3GofskP967Rplgnhm/pW6CxrP0dIeCVtB11llk1AkbACBgBI2AEbgcBr6TdTl+7pUbACBgBI2AEjMCOEPAnOHbUWWbVCMyFgLag2AqKB7s5u4Pj8wR+lf8OC/8agckIWM4mQ3jzBGyk3fwQMAA3igBfXf8ytl1hvm31p65Fv0gf67NvBG4EAcvZjXT0Us30dudSyJquEdg2Ai9kmDUPdH8tdj9Q3NJfY982KubOCMyLgOVsXjxvjppX0m6uy91gI1AhwCraH8bCCBiBRRE4nJzpQc5HJRYdMufEbaSd4+G7jSEghfCLWPqX/EN+kHIM3EFJ8p+Enys86gyZysW/yYksMJnw9znGOSJyg7763/IW+l1YYIxYzi7lwFu4l5gsFuPtzsWgNeGpCEhJoiB/suFwjqTwwDDjO0l/honkPEPhnWiwxcmHRz8qLOrsB0JA48Dy1uhPy1kDjPOgt3DP8Vj0zkbaovCa+FgEwoTR/GPksaQOWU74sOJVGWpTGig6vNnJebSPwqQ0hZzL7hQBy1u64yxnSVwOt4WbbOVGIv0x2410hNl4h4AUI/9J96X8WVZ2RIeVou91YZD8qHuUzCGc2hIP/Bd/fVxlwYOt5AqPcH+Szx9T290IAupvy1tPXwujm5QztbtXdwZsPpPvN8N7xtGYZK+kjUHNZRZDQIIeV3aKjY4cU6L5ly4MPs6YcObmME7tYjXtsXwm2sFO+cGZz27w/3+URxlD67UuuxtBIIwDDBDLW0efC6eblDO1u1N3Br3hoxIdY2dqko20qQi6/NwIYES9kvDPupojevFzE7/OzfAG6DGBcJgXI3So45toYIIfL86acN7N7nYQsLwN7+ublLOc7lR8fKD2UYnhY6g4p9/uLIbMBZZCQELPatAj+d8sUMenoskZt8MZIWrTz7r49wBWRAZt5Sr/ewtgbJI7QkBjwPJW0F83LGcXulNYYKBxVIK0U7jHn/XhuqB7DpvVK2mH7dpdNgwjo/1piLkawqrREVfRIj485bMShvK0MwJDELC8DUHpPM8tytmZ7gw6xkclzsfFYndeSVsMWhMuQUCC/5Xys133qqRcKm9QIkxAPNX9VxfGGWeuJtMWjU06tflXXbzxSbtnO1+0ycaaqckIWN7GQXh0OVP74hZml+7keAS6Gr92KjtoFb8u4MAgBGykDYLJma6AAALOtt2k7UiV56mP7z1xTqJaepcfXxY48koaXYQRynflHuqahCPE7A6NgOVtfPceUs6kMwbpTuXzUYnxY6e4pLc7iyFzgbkRkNCzysUTHMpvtBMdnu4w0Dgr0TwbQZi3lI5uuEQjtOhNz9GAu+AuEbC8Te62w8mZdefkMbEYARtpi0FrwgUIcOj9jRTF1L8kYquPVaT2uTaeEKNiLWBrX1nVboxQ2skqiZ0RyCFgecshMyD+oHJ207pzQLevlsXbnatBP1/FUhqsIKF4caxI4Z4HZXJ3t+3fuYyoL9TMM2MsYAMmcctz20hM546VRL599oGu5mridMqmcBQELG/Te/JocmbdOX1MLELBRtoisF6d6G7/8FaGBFudGJmTjKhgjKXooHxOSj8z3og7qPtR7eLNKybi9oriQZvsZg1FwPI2FKnefIeRM+vO3r5eNYO3O1eFf7bK+fQCk3J0LF2zkoIBtHUX+Z7LiGqvHvEdn2obVXjw1yVxpXHruIziT+1jy5Or+n7RKCIudGQELG8z9O5B5eymdecMw2IREkkj7egT2RQkN4oNZ5D+mNIuyo5p25gyLT4xJjiP1lYQrWzdtyqPYYKhVxthiuPvSjBUIzafTq1HtAY71VXzMrhQyDilrEjQ3j0Y6ElYJrY9SfMokTNgc0h5G4vL2HJhPO1azqJMCIPN6c7IW84f229jy+X4uEb8hZGmRvC9qt0q+CuAxgoVGG3GiZ/vdCFo0WG08XX9wQfxQ5vG9PtUPJ6I12hERf7H+nwf7Clt0cVqIoYfkxL/YkCfsQ14FTcBz8jfFFzp99EGYmRgDX8G3NZg+5p1ThkX8Hk4eZs4ZqbguVs5SwzYzejOBG9nUSv29xkf17q59/bt27ouNZ6VByY5vqpcO92j8GMcQv6ae8UnjQDF8wkAOj0urX+uuJ91n3VKZxJlYo1W/Q99ZbLEFk4IvLL6M/nMj2iALR8FBM856GFoxe+ENQ23LCqqN9nvzQLKQ/+8H+Lom7rvQ1oxHir3UPT+pwsjczdvJIpXtpezfaW0XjwDjp2e6ID5GFyRPwxSVg7n2kbu5HWOxBxuirf+aQA8YVzsRt5CG5P6pgHFSfkmy9oEPHcpZ0389hZes7+XxErtwvbB8fH1D3VxzrzaXbpPLE4RCPBL+dEYi/EoSN4W+zJcH+meCfpP3UcjrMobfxTPpMsKRjTMoJF1yoth8TRkwKDrNeqyxK6QIP74b0nwALOpDmygA66TnPiBFp09+A9vQxsu+r3JiPJwqB9jgbHxShdGYO0UPxaPOC7+rontI4AQJd0QPJMFE5ETcI0rk2NWRhOcLB+Vw03xjBHrn0YXTBgXu5A3ta9T30QocmMmpg/1J+C5OzkbiskW822gvxeBRe1ikeZ3+SzUMJcyz/6icCWv9xu1MrmntoOIP1vlgJji3ug6m6x1X7tQQSVsisxOaqEAq3OVU7m9PPmDVbR+A/flXmjve/LPMC6lFPCmk1k9waBiGT8q5S5yuX6vyohG1Ub51cqRfPod2m0DYAwekb9J59G6GrdQGk87OdeJZ65QR/wYXCOefXLXUe3wJI2FT3RN/YBuDjfiz2RDdVn/3OnqSjaH99Rp8/Kmvq3aJL9P39Ds3JgpgKTOeityVjd4h4G1+3t2yILefCg/LmidFGaO5Z4xebrPT3BfKDG1hcNq2d9KY7Wn6TCmIB4Fv5lGmHLkYcKojeJ0agQAAA/ZSURBVDCFz5zKo9x5nTnmP0vf6k3ACszauBSzLBp0ymin8vQBHcqKw2MuhZnI2Jbuc7l+PwU6cRu6Tees35WXsVOKx6NANBoV7To2d692Mk7rrd4Eg1k8E3l7o8bgqjJxPEV8e+uZmAEZmCoHOdysfxKdM2ZciEwcD5uUN7UJvTVI3wRIcmMmgVh31Bg8dypn3UBsO3XV/l4IGo6FpeaT3xXPw+/D+1SsAPv6OcGtDK3GgKRI0+WUc/zvRJaEzyb0WFg0iadelAd0Jn0rS+Wv7eD9i2tXmqiP5VImM/x4vejos4pET7+T56UuXkCoVzcVjv0dFX5FK/yU4hFpvWkS2XgYI7jGo8nrADyb2UvCpbhG2hHfeL9Jvwc36598r5WOizgetipvg/VNz5jJI9adUopnpBZxjff2Z0ZgY/09Z+uYt1OLKYxF3CcP7vzqDbjcxIOll3I89ZwEXsoKbObnvFFuEPPdqm90xa2SJA9NYkuFxQPteaaLSZgtQwzIuK3CSmJqlRF+OXuXSlN0hQ906QgUY8ThmehhxFIHW8b4v+q+wlo+Zb4P8awyssyLIY3j7B771+xdV07hsX94C+9JzEUTXqkTDJourorGQdRM68WjmVnheDA4NUhbWde7Df3B2GCbk7EBM2wlsr1Mv0aXxTNmUH76kvEex0JMij5nHdvjqRRXaMHXo0h0434WN2Fh/XO3k9EeE3Rp6bjYrLypn0v1TXbMAIzoWc4AYsNOfVQy526lv2dDNIz5PnqPopHGxFvtf/aVID2AG42YiyJKJ40VHVw1mROnq57YFcYAiIqHDjgprs/gI9tSDsOJSZcDe2DBuS4mTISdK/LarB8DFAMs6VQWxcNbGlX7YiboEpYPHhhrEasqi+7BgXjoM9FiOFZGmXwGNi9t/KyrxlNxY1xXv+cMZ+rHperuxOOu2NlvzlA5y7TmjTCu5EJ+7LOTwjxYVGNcPkZbxKILz5PyVeNK7cHwwDBlXJ0ZwcqTkoFSXEW2or95fGFUrhO3uyzvfoURY9D6R8abcMjqn3eI1aEtj4dSfZMdM5azur+3HiiZc1ft7zCmSmQN7P9SudxDJunM7bg3d17y92E00hBeJo2hjtUfjIR6NadVkMbEFZo4gaFYq7DKVcpCfmSumb9Favlb8YHCR+HhAA6DMh66hteUgUZeMKNszjGwntDeRlvJ2zaIc9iD1ycqW3e0wnQ8NMAsxxfpQ1xXv7NyhOPNzyoQfj6TjwEb+7WZ1odHMy/hapCKVhwH7fRV78UX4/y1/Gig1eOU9utijGNoxf7J4qm8jH8eAJoHRE+6/0vxfa4U10ivwjfebNjP4pbh2frnDpjScbFleSvVN8kxYznLSMzGotVPpXPuqv0tfs9eXroinO8/CJUhvG+GVCxmMTCYoOLElCrGKlBlQMiPRkXTmHmh+LgyRDwdMPg8msqS/7dQTt4gx2c9chMi9H4MVJhM6w5RmTqcqAVDhbJJp7JsYaJI/0dYPm3sMm5TdDjTl3LZelOZM3Fd/Q4O9RYs5dUG6sRIe8V9wnXikcg/Kkp8vPu43ygK+UKifY9U+TzZY5T9H/fBsWrWXPl6o3jyRJfFU+UYe/X40z34Un6IuwqufYyIZ2S/2d5YhHaflJ6Slb6nSYpmcSOx6QIPq+of+BEfyMJcOghas+sf+JzDqa2Ly5v4RB5K9E1yzIhXy1lHpy/Zlx3V1kmqv9Kviigd80fsb2yDnKO9uP9GI+3utudXADNxPZJ/tn3XU4xkJqNqZUplmeTr1QSFo9KPK2+K6naiAb3J3xWLtYheNXnKj7xEhRmz5PzkwGllhs+Xumh3tXWqejb98VbxhwDh2oZzfEkit4I3BI87yne/1SClPl306SCnvFHQB+UfmQlj7Mcevh4rT2pFcUiVrBy08c2VK8U10ulSAjHPYF9YpIywk+IZ26w+Vw9egwkWZhT9Tegf2BYvs+kg0VpS/zRR3qS8qf1j9U2zbbmw5ayBjLC+hu5s1JgOThjzaYLvYq/R3+9qmxASBuxIQSGO/ya1GPdPNNIQ3hjZzFiHRQxF/KH8egVN4Wp1TH49USnMxHV2xkr3pKPEqzqa+RWPwQezlaJSeE1H23hKfzOQCdpTKb5UftEBH7bLmPC5Tgoz0fCpDK6129zX723+aAMGZg6fTjxof8tFOmONkBa5WW/pu3ocq820LW6Jn8I9Rn0tDwr34akstUOemmXrhESgFFdIgGktlwmaW4rqxU14W/9c9ljpuNiyvNG6En3TO2YCXJazy3GzpZihc+6q/S39k9tF6MJyyC4Ci1OVHdUihP7G/RqNtMqIuou7/BWDGF5P5VeGRiMHAtBeVWHiaq6UkR36xL8QjfbTNvGDV9GUd0mX4r2rvr6JENzogLrNan/8N4Ynim8rpa66lkhL9rt4jBZ+Xafi6Gva2x4DdZ6QXmIY/DcUZrLZmqNvmny1x+n3Sqcvm2M9iWe7YSrDmOChZWj/942zdhXxvqQvYpk1/E7chBNyZP1z2TOl42KT8qb+HaNvOscMUImu5exyzGwtZuicu2p/aywldxFmAJPzteywtR07cBh5bx6EFCaLp+1c3IeBDiHOC2BNNt0niqsNkJDAG5HtuN+VxiTfnNBOyofyZSIcuu2jrMs48VIJtKiX8FIB2cMRB+/bq0+0uW2YNg2CSBIl/DreLOBn+1110VcVT+Ifn4H0scJv5OfcEDyaZbtoNfOtEX6uShn3cSxjJFTjVz7xrJC2BbcLz2YbkIUSA6oUV+qiz5YcO9Qxl8viJoyRS+ufNNKl42LL8laqb7JjpgGV5awBxtJBySo6h6M9OOQW91zxyXEXZJt8Q+bcrfR31ai5foQBtgFfleBzZHF+AUeOFn1MPdFI+0FhFGHKASBAsk3XdgBXOVVAeYwuVgjYJmI7L66ykQ9GqolJPtYzExz5caSx7Znt0CrXsj+0Mb6xN7Sm2I5cfgYnGLCCGPO8r0D156mKo/0M6ick6h4MMQ7ghXjSecok/pUu4qNhgPGH4TB0y0xFL1xXv8MHfFIn5wmbn5q4IBQi+vBol3sdIh61E9a+V7urpWr5PJiwzclnUDBUEaBXpMtvuy48m3l5IILuUFeEq3iDRxzjbw+uCzfrn3wPFo0LkdmsvIm3Un3TNWYiYpaziMR1fOaLOD+dFEbHYQvELyW0uSiZc1fv7zbzM97zsAV2jFdWu/FZEKnmmHtv3969uKMIJqLPY4LCdh0ICCcGGJ9UyA3AjtLbSZqr38fgoTIYoQhx6gOuo0AKNL9XYfqHg/+10hhFUIVEA1qsGseHjiwp5ZlVjkLdReNMZSKuPPzElcAsz1MTVAcrFpNeHBCNWXGb2qatlxdexfpHZeK42LS8DcV+zjEzEc/dyFkXtmF8jNadKo8xwcN8tUsUMEWu+dpD6qG2i52LNNGYTUcE3or06gVDV4q436jna4UnT2gNekcPcjYLzPbu5ur3YjwagsuTxCwOmrqgx2oSqzBzOCa39vZ0ju5ceEb6xbiqIBM4rmRL9a7EuF9W7Kau2s2N27iW7KdU8bjYkbwN7YU5x0wxnmJyj3KWxXYG3Yn98Ee2gukJa/f39BaMoFAbaeogVgl4Go4DbwS52ygSMAKr3pWVrSMyR79PxANDotrunQsr8cM2EG6oYXWXO//LtvIgg2cOPCMbE3CNMjz56TXy0uWLT86rTpKFOXHr4vUIaRPGBc3fg7wN6qa5xswEPHcnZ33ACovRupP+0NV8WMNo4wjRLHoI+qI32UYRHfptN/P3/Vanfa77krMyreI3cwtGR1p1nNrvU/BAgKOym2sAcb6x5FMqffVyTqDETcUz1jUWV840nKSMBhmWsbIN+HPhtoGmLMrC2HEBU3uQtxLw5hgzY/Hcq5x14TuL7pTuYfeBYxCz7ZIEptfs7y7cFks7M9IELFYw5xW+WqzGnRMO2IDR3ibALPJT+n0GPNiSfCg6D7MMlifwNDjXKtpJvBWd61L+yXIkGsjg2HE2a/vL4R9XYg7cxtW8n1ITxwUN3by8lfTG1DEzEc9dylkPvpPbJEx56GZrkrNozZW1nqr7k1fu734GF8hRvziwAG2TNAK9CEjo4mFmXlqpXkHuLdTIoDJRIWA0s+KFccbLCGf0lA+lwRvKOWNwtsPUqmM1F/DggO1VDjOv1lBXPAoBjQ/L2yjkzgsdQc5CG9CLs+nOQBPdU+00hfuT/MMsapyPhOXvHixfhWswAnkEJLzVB/uU45muIiNNZXnq4/MkPLFVSkB+fFmgXklTHNsZPNGxVP5aF4qJg8K1U55Zzk3UBNcLgAmuCMu7Iv49OgKMc13IguVtWmfvWs40BmbXnaLJAzO6FiONhwEcxtqZrq1i/TMYgQeDczqjEVgOgR9FOv4v6KBapARYEcNAQyE0n9IIx4noFJQFr1rXRovCxB/FKGvjFc+UNDFp5/H9bSNgeZve/7uVM+m+pXQnOxjQxq+d6jvS+e26XdcK2Ei7FtKupwsBjK0XEma+RVavgHUVUBqrYZxla79VyBNi0yDDGKsNMuXnCY+VhKM6Duv6yfWovTtPuyxv03Hcs5wtojulW9+bDqsptBG4347wvRG4NgLBMGPlh+3IoY6VtzODTnR4imPJPW55pmixzdOVniqzizi1P/4rSNtw3QX/ZvI6CFjepuF8ADmz7pw2BK5a2kbaVeF2ZR0I8HTHahqGVqcLecjXNraqLdMwCeVo8AT8Ry5x5/GsoP2s9h95pXDnXbQZ9i1v47tit3Jm3Tm+09cqaSNtLeRd7xkCUh6s/mBcvDxL6L5pn7vinEi1tSl6/GEtq2q1C/d8xLDe/qwTdx5Qm9jmpb3e6tx5X16DfY0Xy9sIoA8kZ9adI/p/jSI20tZA3XXmEMDA+EqKsHM1TekYc2x11kaY4lgh47xZXCVL/SE8edrKSVGHcPHp/qjtO0QnbawRlrfyDtm1nFl3lnf42iVspK3dA66/RkAKhKd7VrmGrKZxfo2/a8KoY+sG44SVtEfEyedV8LZ7qohUfDvfru7VXlbRuJ7vinEzuyoCGjeWt4IeOJCcWXcW9PvaWf0x27V7wPWfISBFyGrYb7rqb5+dZfDNBQLCLH68tn6r9SKTI4xAAgHLWwKUTJTlLAOMoxdFwCtpi8Jr4qUISBGykvZK1+FWvEqxGJJfeLGKyHfhbKANAcx5zhCwvJ3Bkb2xnGWhccLCCNhIWxhgky9HQAqR/8p8ExRjOYEbKSF8qm1O+SWfLrkRdNzMoQhY3rqRspx14+PUZRGwkbYsvqY+EoFgePBxWw7727UQEC5sC7OK9nErybdGoBgBy1saMstZGhfHXg8BG2nXw9o1FSIgBfmRivDH5xgkdgEB4cHbr9/r+lhh3nS1MwKTEbC8nUNoOTvHw3frIPD/AQCcRy9bh963AAAAAElFTkSuQmCC",
      "text/latex": [
       "$\\displaystyle - \\frac{M \\left(2 M - r\\right) \\left(\\frac{d}{d \\tau} t{\\left(\\tau \\right)}\\right)^{2}}{r^{3}} + \\frac{M \\left(2 M - r\\right) \\left(\\frac{d}{d \\tau} r{\\left(\\tau \\right)}\\right)^{2}}{r^{3} \\left(- \\frac{2 M}{r} + 1\\right)^{2}} + \\left(2 M - r\\right) \\sin^{2}{\\left(\\theta \\right)} \\left(\\frac{d}{d \\tau} \\phi{\\left(\\tau \\right)}\\right)^{2} + \\left(2 M - r\\right) \\left(\\frac{d}{d \\tau} \\theta{\\left(\\tau \\right)}\\right)^{2} + \\frac{d^{2}}{d \\tau^{2}} r{\\left(\\tau \\right)} = 0$"
      ],
      "text/plain": [
       "                        2                         2                            ↪\n",
       "              ⎛d       ⎞                ⎛d       ⎞                             ↪\n",
       "  M⋅(2⋅M - r)⋅⎜──(t(τ))⎟    M⋅(2⋅M - r)⋅⎜──(r(τ))⎟                             ↪\n",
       "              ⎝dτ      ⎠                ⎝dτ      ⎠                 2    ⎛d     ↪\n",
       "- ─────────────────────── + ─────────────────────── + (2⋅M - r)⋅sin (θ)⋅⎜──(φ( ↪\n",
       "             3                                2                         ⎝dτ    ↪\n",
       "            r                    3 ⎛  2⋅M    ⎞                                 ↪\n",
       "                                r ⋅⎜- ─── + 1⎟                                 ↪\n",
       "                                   ⎝   r     ⎠                                 ↪\n",
       "\n",
       "↪                                              \n",
       "↪                                              \n",
       "↪     2                       2    2           \n",
       "↪    ⎞              ⎛d       ⎞    d            \n",
       "↪ τ))⎟  + (2⋅M - r)⋅⎜──(θ(τ))⎟  + ───(r(τ)) = 0\n",
       "↪    ⎠              ⎝dτ      ⎠      2          \n",
       "↪                                 dτ           \n",
       "↪                                              \n",
       "↪                                              "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle - \\sin{\\left(\\theta \\right)} \\cos{\\left(\\theta \\right)} \\left(\\frac{d}{d \\tau} \\phi{\\left(\\tau \\right)}\\right)^{2} + \\frac{d^{2}}{d \\tau^{2}} \\theta{\\left(\\tau \\right)} + \\frac{2 \\frac{d}{d \\tau} r{\\left(\\tau \\right)} \\frac{d}{d \\tau} \\theta{\\left(\\tau \\right)}}{r} = 0$"
      ],
      "text/plain": [
       "                                            d        d           \n",
       "                          2    2          2⋅──(r(τ))⋅──(θ(τ))    \n",
       "                ⎛d       ⎞    d             dτ       dτ          \n",
       "- sin(θ)⋅cos(θ)⋅⎜──(φ(τ))⎟  + ───(θ(τ)) + ─────────────────── = 0\n",
       "                ⎝dτ      ⎠      2                  r             \n",
       "                              dτ                                 "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\tau^{2}} \\phi{\\left(\\tau \\right)} + \\frac{2 \\cos{\\left(\\theta \\right)} \\frac{d}{d \\tau} \\phi{\\left(\\tau \\right)} \\frac{d}{d \\tau} \\theta{\\left(\\tau \\right)}}{\\sin{\\left(\\theta \\right)}} + \\frac{2 \\frac{d}{d \\tau} \\phi{\\left(\\tau \\right)} \\frac{d}{d \\tau} r{\\left(\\tau \\right)}}{r} = 0$"
      ],
      "text/plain": [
       "                     d        d            d        d           \n",
       " 2          2⋅cos(θ)⋅──(φ(τ))⋅──(θ(τ))   2⋅──(φ(τ))⋅──(r(τ))    \n",
       "d                    dτ       dτ           dτ       dτ          \n",
       "───(φ(τ)) + ────────────────────────── + ─────────────────── = 0\n",
       "  2                   sin(θ)                      r             \n",
       "dτ                                                              "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Beispiel einer gebundenen elliptischen Bahn\n",
    "\n",
    "Wir lassen im Folgenden nur ebene Bewegungen zu ($\\theta=\\frac{\\pi}{2}, {\\rm cos}(\\theta)=0, {\\rm sin}(\\theta)=1, \\frac{d \\theta}{d\\tau}=0, \\frac{d^2 \\theta}{d\\tau^2}=0$). Unter diesen Annahmen sind nur drei der vier Geodätengleichungen relevant und diese lauten dann:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "GeodEq0 = GeodEq0.subs({(theta,pi/2),(diff(x2,tau),0)})\n",
    "GeodEq1 = GeodEq1.subs({(theta,pi/2),(diff(x2,tau),0)})\n",
    "GeodEq3 = GeodEq3.subs({(theta,pi/2),(diff(x2,tau),0)})\n",
    "\n",
    "GeodEq0_a = Eq(diff(x0,tau,tau),solve(GeodEq0,diff(x0,tau,tau))[0])\n",
    "GeodEq1_a = Eq(diff(x1,tau,tau),solve(GeodEq1,diff(x1,tau,tau))[0])\n",
    "GeodEq3_a = Eq(diff(x3,tau,tau),solve(GeodEq3,diff(x3,tau,tau))[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\tau^{2}} t{\\left(\\tau \\right)} = \\frac{2 M \\frac{d}{d \\tau} r{\\left(\\tau \\right)} \\frac{d}{d \\tau} t{\\left(\\tau \\right)}}{r \\left(2 M - r\\right)}$"
      ],
      "text/plain": [
       "                d        d       \n",
       " 2          2⋅M⋅──(r(τ))⋅──(t(τ))\n",
       "d               dτ       dτ      \n",
       "───(t(τ)) = ─────────────────────\n",
       "  2              r⋅(2⋅M - r)     \n",
       "dτ                               "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq0_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\tau^{2}} r{\\left(\\tau \\right)} = - \\frac{2 M^{2} \\left(\\frac{d}{d \\tau} r{\\left(\\tau \\right)}\\right)^{2}}{r \\left(2 M - r\\right)^{2}} + \\frac{2 M^{2} \\left(\\frac{d}{d \\tau} t{\\left(\\tau \\right)}\\right)^{2}}{r^{3}} + \\frac{M \\left(\\frac{d}{d \\tau} r{\\left(\\tau \\right)}\\right)^{2}}{\\left(2 M - r\\right)^{2}} - \\frac{M \\left(\\frac{d}{d \\tau} t{\\left(\\tau \\right)}\\right)^{2}}{r^{2}} + \\frac{\\left(- 2 M + r\\right)^{3} \\left(\\frac{d}{d \\tau} \\phi{\\left(\\tau \\right)}\\right)^{2}}{\\left(2 M - r\\right)^{2}}$"
      ],
      "text/plain": [
       "                             2                  2               2              ↪\n",
       "                 2 ⎛d       ⎞       2 ⎛d       ⎞      ⎛d       ⎞      ⎛d       ↪\n",
       " 2            2⋅M ⋅⎜──(r(τ))⎟    2⋅M ⋅⎜──(t(τ))⎟    M⋅⎜──(r(τ))⎟    M⋅⎜──(t(τ) ↪\n",
       "d                  ⎝dτ      ⎠         ⎝dτ      ⎠      ⎝dτ      ⎠      ⎝dτ      ↪\n",
       "───(r(τ)) = - ──────────────── + ──────────────── + ───────────── - ────────── ↪\n",
       "  2                        2             3                    2           2    ↪\n",
       "dτ              r⋅(2⋅M - r)             r            (2⋅M - r)           r     ↪\n",
       "\n",
       "↪   2                         2\n",
       "↪  ⎞              3 ⎛d       ⎞ \n",
       "↪ )⎟    (-2⋅M + r) ⋅⎜──(φ(τ))⎟ \n",
       "↪  ⎠                ⎝dτ      ⎠ \n",
       "↪ ─── + ───────────────────────\n",
       "↪                      2       \n",
       "↪             (2⋅M - r)        "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq1_a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\tau^{2}} \\phi{\\left(\\tau \\right)} = - \\frac{2 \\frac{d}{d \\tau} \\phi{\\left(\\tau \\right)} \\frac{d}{d \\tau} r{\\left(\\tau \\right)}}{r}$"
      ],
      "text/plain": [
       "               d        d        \n",
       " 2          -2⋅──(φ(τ))⋅──(r(τ)) \n",
       "d              dτ       dτ       \n",
       "───(φ(τ)) = ─────────────────────\n",
       "  2                   r          \n",
       "dτ                               "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "GeodEq3_a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Um dieses System von drei gekoppelten Differenzialgleichungen zweiter Ordnung mit Python nummerisch lösen zu können, müssen wir es zunächst in ein System von sechs Differenzialgleichungen erster Ordnung umschreiben. Details zur numerischen Lösung von Differenzialgleichungen finden Sie in der Vorlesung <A href=\"https://itp.uni-frankfurt.de/~hanauske/VPROG/index.html\" target=\"_blank\">Einführung in die Programmierung für Studierende der Physik</A> in den Unterpunkten <A href=\"https://itp.uni-frankfurt.de/~hanauske/VPROG/V8/DGL_1.html\" target=\"_blank\">Differenzialgleichungen: Numerische Lösung von Anfangswertproblemen</A> und \n",
    "<A href=\"https://itp.uni-frankfurt.de/~hanauske/VPROG/V9/DGL_2.html\" target=\"_blank\">Systeme von gekoppelten Differenzialgleichungen und Differenzialgleichungen zweiter Ordnung</A>.\n",
    "\n",
    "Wir definieren dazu formal\n",
    "\n",
    "$$\n",
    "t(\\tau) = y_1(\\tau) \\, , \\quad r(\\tau) = y_2(\\tau), \\quad \\phi(\\tau) = y_3(\\tau) \\, , \\quad \\frac{d t(\\tau)}{d\\tau} = y_4(\\tau) \\, , \\quad \\frac{d r(\\tau)}{d\\tau} = y_5(\\tau), \\quad \\frac{d \\phi(\\tau)}{d\\tau} = y_6\n",
    "$$\n",
    "\n",
    "und schreiben unser DGL-System wie folgt um:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "y1, y2, y3, y4, y5, y6 = symbols('y_1, y_2, y_3, y_4, y_5, y_6')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\tau^{2}} t{\\left(\\tau \\right)} = \\frac{2 M y_{4} y_{5}}{y_{2} \\left(2 M - y_{2}\\right)}$"
      ],
      "text/plain": [
       " 2                       \n",
       "d             2⋅M⋅y₄⋅y₅  \n",
       "───(t(τ)) = ─────────────\n",
       "  2         y₂⋅(2⋅M - y₂)\n",
       "dτ                       "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SysDGL_0 = Eq(diff(x0,tau,tau),GeodEq0_a.rhs.subs({(r,y2),(diff(x0,tau),y4),(diff(x1,tau),y5),(diff(x3,tau),y6)}))\n",
    "SysDGL_0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\tau^{2}} r{\\left(\\tau \\right)} = - \\frac{2 M^{2} y_{5}^{2}}{y_{2} \\left(2 M - y_{2}\\right)^{2}} + \\frac{2 M^{2} y_{4}^{2}}{y_{2}^{3}} + \\frac{M y_{5}^{2}}{\\left(2 M - y_{2}\\right)^{2}} - \\frac{M y_{4}^{2}}{y_{2}^{2}} + \\frac{y_{6}^{2} \\left(- 2 M + y_{2}\\right)^{3}}{\\left(2 M - y_{2}\\right)^{2}}$"
      ],
      "text/plain": [
       " 2                  2   2         2   2          2          2     2            3\n",
       "d                2⋅M ⋅y₅       2⋅M ⋅y₄       M⋅y₅       M⋅y₄    y₆ ⋅(-2⋅M + y₂) \n",
       "───(r(τ)) = - ────────────── + ──────── + ─────────── - ───── + ────────────────\n",
       "  2                        2       3                2      2                2   \n",
       "dτ            y₂⋅(2⋅M - y₂)      y₂       (2⋅M - y₂)     y₂       (2⋅M - y₂)    "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SysDGL_1 = Eq(diff(x1,tau,tau),GeodEq1_a.rhs.subs({(r,y2),(diff(x0,tau),y4),(diff(x1,tau),y5),(diff(x3,tau),y6)}))\n",
    "SysDGL_1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{d^{2}}{d \\tau^{2}} \\phi{\\left(\\tau \\right)} = - \\frac{2 y_{5} y_{6}}{y_{2}}$"
      ],
      "text/plain": [
       " 2                   \n",
       "d           -2⋅y₅⋅y₆ \n",
       "───(φ(τ)) = ─────────\n",
       "  2            y₂    \n",
       "dτ                   "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SysDGL_3 = Eq(diff(x3,tau,tau),GeodEq3_a.rhs.subs({(r,y2),(diff(x0,tau),y4),(diff(x1,tau),y5),(diff(x3,tau),y6)}))\n",
    "SysDGL_3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Dieses System von drei gekoppelten DGLs 2. Ordnung in der Eigenzeit $\\tau$ werden wir später zusätzlich mittels der Programmiersprache C++ lösen. Wir exportieren deshalb schon jetzt das zugrundeliegende System von DGLs als C++ Code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "wert = 2*M*y_4*y_5/(y_2*(2*M - y_2));\n",
      "wert = -2*pow(M, 2)*pow(y_5, 2)/(y_2*pow(2*M - y_2, 2)) + 2*pow(M, 2)*pow(y_4, 2)/pow(y_2, 3) + M*pow(y_5, 2)/pow(2*M - y_2, 2) - M*pow(y_4, 2)/pow(y_2, 2) + pow(y_6, 2)*pow(-2*M + y_2, 3)/pow(2*M - y_2, 2);\n",
      "wert = -2*y_5*y_6/y_2;\n"
     ]
    }
   ],
   "source": [
    "from sympy.codegen.ast import Assignment\n",
    "\n",
    "wert = symbols('wert')\n",
    "print(ccode(Assignment(wert,SysDGL_0.rhs)))\n",
    "print(ccode(Assignment(wert,SysDGL_1.rhs)))\n",
    "print(ccode(Assignment(wert,SysDGL_3.rhs)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nun lösen wir die DGLs mittels Python und setzen im Folgenden $M=1$. Wir benutzen hierfür das Python-Modul <A href=\"https://docs.scipy.org/doc/scipy/tutorial/index.html#user-guide\" target=\"_blank\">SciPy</A>, welches eine breite Kollektion von mathematischen Algorithmen und Funktionen bereitstellt. Im Speziellen werden wir in diesem Semester die Funktion <A href=\"https://docs.scipy.org/doc/scipy/reference/generated/scipy.integrate.solve_ivp.html#scipy.integrate.solve_ivp\" target=\"_blank\">'solve_ivp(...)'</A> verwenden, die sich im Untermodul <A href=\"https://docs.scipy.org/doc/scipy/reference/integrate.html#module-scipy.integrate\" target=\"_blank\">'scipy.integrate'</A> befindet, welches Funktionen zum Lösen von gewöhnlichen Differenzialgleichungen bereitstellt. Zusätzlich berechnen wir die numerische Lösung auch mittels der älteren Methode \"integrate.odeint()\", die ebenfalls in dem Python-Modul \"scipy\" definiert ist. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt \n",
    "from matplotlib import rcParams\n",
    "from scipy import integrate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir definieren das System der gekoppelten sechs Differenzialgleichungen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "set_M = 1\n",
    "f_4 = lambdify((y1,y2,y3,y4,y5,y6), SysDGL_0.rhs.subs(M,set_M))\n",
    "f_5 = lambdify((y1,y2,y3,y4,y5,y6), SysDGL_1.rhs.subs(M,set_M))\n",
    "f_6 = lambdify((y1,y2,y3,y4,y5,y6), SysDGL_3.rhs.subs(M,set_M))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def DGLsys(vy, time):\n",
    "    y1,y2,y3,y4,y5,y6 = vy\n",
    "    dy1 = y4\n",
    "    dy2 = y5\n",
    "    dy3 = y6\n",
    "    dy4 = f_4(y1,y2,y3,y4,y5,y6)\n",
    "    dy5 = f_5(y1,y2,y3,y4,y5,y6)\n",
    "    dy6 = f_6(y1,y2,y3,y4,y5,y6)\n",
    "    return np.array([dy1,dy2,dy3,dy4,dy5,dy6])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def DGLsys_ivp(time, vy):\n",
    "    y1,y2,y3,y4,y5,y6 = vy\n",
    "    dy1 = y4\n",
    "    dy2 = y5\n",
    "    dy3 = y6\n",
    "    dy4 = f_4(y1,y2,y3,y4,y5,y6)\n",
    "    dy5 = f_5(y1,y2,y3,y4,y5,y6)\n",
    "    dy6 = f_6(y1,y2,y3,y4,y5,y6)\n",
    "    return np.array([dy1,dy2,dy3,dy4,dy5,dy6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In Abhängigkeit von den Anfangswerten können unterschiedliche Bahnen der Bewegung entstehen. Wir wählen zunächst als Beispiel einer gebundenen elliptischen Bahn. Zur Zeit $t=\\tau=0$ befindet sich der Probekörper bei einem Radius von $r_0=10=5 R_S$ und Winkel $\\phi_0=0$. Seine Anfangsgeschwindigkeit in radialer Richtung sei $\\left. \\frac{dr}{d\\tau} \\right|_{\\tau=0}=0$ und seine Anfangsgeschwindigkeit in $\\phi$-Richtung sei $\\left. \\frac{dr}{d\\tau} \\right|_{\\tau=0}=0.041$. Die weitere nötige Anfangsbedingung ( $\\left. \\frac{dt}{d\\tau} \\right|_{\\tau=0}$ ) erhalten wir mittels der Bedingung $\\left( \\frac{ds}{d\\tau} \\right)^2= g_{\\mu\\nu} \\frac{dx^\\mu}{d\\tau} \\frac{dx^\\nu}{d\\tau} = g_{\\mu\\nu} u^\\mu u^\\nu = 1$, die für massive Probekörper immer erfüllt sein muss ($u^\\mu$ ist die Vierergeschwindigkeit des Probekörpers). Die Gleichung für das infinitesimale Weglängenelement $ds^2=1$ in der Ebene lautet:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle - d\\phi^{2} r^{2} - \\frac{dr^{2}}{- \\frac{2 M}{r} + 1} + dt^{2} \\left(- \\frac{2 M}{r} + 1\\right) = 1$"
      ],
      "text/plain": [
       "                   2                         \n",
       "       2  2      dr         2 ⎛  2⋅M    ⎞    \n",
       "- d\\phi ⋅r  - ───────── + dt ⋅⎜- ─── + 1⎟ = 1\n",
       "                2⋅M           ⎝   r     ⎠    \n",
       "              - ─── + 1                      \n",
       "                 r                           "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dt, dr, dtheta, dphi = symbols('dt, dr, d\\\\theta, d\\\\phi')\n",
    "dx = GenericVector([dt, dr, dtheta, dphi],[t, r, theta, phi], config='u',parent_metric=Metric)\n",
    "dxdx = tensorproduct(dx.tensor(),dx.tensor())\n",
    "dxdxT = Tensor(dxdx,config=\"uu\")\n",
    "ds2a = tensorproduct(g.tensor(),dxdxT.tensor())\n",
    "ds2aT = Tensor(ds2a,config=\"lluu\")\n",
    "ds2 = tensorcontraction(ds2aT.tensor(), (0, 2),(1, 3))\n",
    "Eq_ds = Eq(ds2.subs({(theta,pi/2),(dtheta,0)}),1)\n",
    "Eq_ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir lösen diese Gleichung nach $dt$ auf:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\sqrt{r \\left(- 2 M d\\phi^{2} r^{2} - 2 M + d\\phi^{2} r^{3} + dr^{2} r + r\\right)}}{- 2 M + r}, \\  \\frac{\\sqrt{r \\left(- 2 M d\\phi^{2} r^{2} - 2 M + d\\phi^{2} r^{3} + dr^{2} r + r\\right)}}{2 M - r}\\right]$"
      ],
      "text/plain": [
       "⎡   ___________________________________________________     __________________ ↪\n",
       "⎢  ╱   ⎛           2  2              2  3     2      ⎞     ╱   ⎛           2   ↪\n",
       "⎢╲╱  r⋅⎝- 2⋅M⋅d\\phi ⋅r  - 2⋅M + d\\phi ⋅r  + dr ⋅r + r⎠   ╲╱  r⋅⎝- 2⋅M⋅d\\phi ⋅r ↪\n",
       "⎢──────────────────────────────────────────────────────, ───────────────────── ↪\n",
       "⎣                       -2⋅M + r                                               ↪\n",
       "\n",
       "↪ _________________________________⎤\n",
       "↪ 2              2  3     2      ⎞ ⎥\n",
       "↪   - 2⋅M + d\\phi ⋅r  + dr ⋅r + r⎠ ⎥\n",
       "↪ ─────────────────────────────────⎥\n",
       "↪   2⋅M - r                        ⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LoesEq_ds = solve(Eq_ds,dt)\n",
    "LoesEq_ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir benutzen die erste Lösung (positive Lösung) und berechnen für unsere Anfangsbedingung $\\left. \\frac{dt}{d\\tau} \\right|_{\\tau=0}=1.20835632162041$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 100000\n",
    "tauval = np.linspace(0, 800, N+1)\n",
    "\n",
    "r0 = 10.0\n",
    "t0 = 0.0\n",
    "phi0 = 0.0\n",
    "dr0 = 0.0\n",
    "dphi0 = 0.041\n",
    "dt0 = float(LoesEq_ds[0].subs({(r,r0),(dr,dr0),(dphi,dphi0),(M,set_M)}))\n",
    "\n",
    "initialval = np.array([t0,r0,phi0,dt0,dr0,dphi0])\n",
    "\n",
    "Loes = integrate.odeint(DGLsys, initialval, tauval)\n",
    "Loes_ivp = integrate.solve_ivp(DGLsys_ivp, [tauval[0], tauval[-1]], initialval, t_eval=tauval, rtol=10**(-10), atol=10**(-10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "dt0 = 1.208356321620407\n"
     ]
    }
   ],
   "source": [
    "print(\"dt0 =\",dt0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "rcParams.update({\n",
    "    'figure.figsize'    : [6,6],\n",
    "    'axes.titlesize' : 14,\n",
    "    'axes.labelsize' : 16,  \n",
    "    'xtick.labelsize' : 14 ,\n",
    "    'ytick.labelsize' : 14 \n",
    "})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.cla()\n",
    "plt.ylabel(r\"$\\rm Radius \\,\\,r \\,[km]$\")\n",
    "plt.xlabel(r\"$\\rm Eigenzeit \\,\\,\\tau$\")\n",
    "plt.plot(tauval,Loes[:, 1],c=\"blue\", linewidth=1.5, linestyle='-');\n",
    "plt.plot(Loes_ivp.t,Loes_ivp.y[1],c=\"red\", linewidth=2, linestyle=':');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Das obere Diagramm zeigt den Abstand des Probekörpers vom schwarzen Loch (Radius r) als Funktion der Eigenzeit $\\tau$, wobei die blaue Kurve die Ergebnisse der odeint-Methode und die gepunktete rote Kurve die solve_ivp-Simulation zeigt. Beide Simulationen stimmen gut miteinander überein. Am Ende der Simulation ergeben sich die folgenden Werte für $(t,r,\\phi,\\frac{d t(\\tau)}{d\\tau},\\frac{d r(\\tau)}{d\\tau},\\frac{d \\phi(\\tau)}{d\\tau})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "solve_ivp-Methode:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "908.6596297139737 11.03794693307119 18.670947066104457 1.180602015359073 0.05196781395818743 0.03365171895832977\n"
     ]
    }
   ],
   "source": [
    "print(Loes_ivp.y[0][-1],Loes_ivp.y[1][-1],Loes_ivp.y[2][-1],Loes_ivp.y[3][-1],Loes_ivp.y[4][-1],Loes_ivp.y[5][-1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "odeint-Methode:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "908.6598050881037 11.037902200863277 18.670989940158336 1.1806030235883063 0.05196848574304601 0.033651962768271884\n"
     ]
    }
   ],
   "source": [
    "print(Loes[-1, 0],Loes[-1, 1],Loes[-1, 2],Loes[-1, 3],Loes[-1, 4],Loes[-1, 5])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Trajektorie des Körpers wird in dem unteren Bild dargestellt, wobei der rote Punkt den Anfangsort des Probekörpers zeigt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = plt.gca()\n",
    "ax.cla() \n",
    "ax.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax.plot(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]),c=\"blue\", linewidth=1.5, linestyle='-');\n",
    "ax.scatter(Loes[0, 1]*np.cos(Loes[0, 2]),Loes[0, 1]*np.sin(Loes[0, 2]),c=\"red\", s=60, marker='o');\n",
    "BH = plt.Circle((0, 0), 2, color='black')\n",
    "ax.add_patch(BH);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir werden in Kürze sehen, dass es Erhaltungsgrößen während der Bewegung des Probekörpers gibt. Während der gesamten Bewegung wird der Drehimpuls $l$ pro Masse $m$, die Teilchenenergie $e$ pro Masse und das Weglängenelement $ds^2$ erhalten. In der numerischen Simulation sind diese Größen ebenfalls in Näherung erhalten, wie man in den folgenden Abbildungen sieht."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.gridspec as gridspec"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1600x500 with 3 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(16,5))\n",
    "gs = gridspec.GridSpec(1, 3, width_ratios=[1,1,1], wspace=0.25)\n",
    "ax1 = plt.subplot(gs[0])\n",
    "ax2 = plt.subplot(gs[1])\n",
    "ax3 = plt.subplot(gs[2])\n",
    "\n",
    "ax1.set_ylabel(r\"$\\rm Drehimpuls \\,\\,l$\")\n",
    "ax1.set_xlabel(r\"$\\rm Eigenzeit \\,\\,\\tau$\")\n",
    "ax2.set_ylabel(r\"$\\rm Energie \\,\\,e $\")\n",
    "ax2.set_xlabel(r\"$\\rm Eigenzeit \\,\\,\\tau$\")\n",
    "ax3.set_ylabel(r\"$\\rm Weglaengenelement \\,\\,ds^2$\")\n",
    "ax3.set_xlabel(r\"$\\rm Eigenzeit \\,\\,\\tau$\")\n",
    "\n",
    "data_l = Loes[:, 1]**2*Loes[:, 5]\n",
    "data_E = (1-2/Loes[:, 1])*Loes[:, 3]\n",
    "data_ds2 = (1-2/Loes[:, 1])*Loes[:, 3]**2-Loes[:, 4]**2/(1-2/Loes[:, 1])-Loes[:, 5]**2*Loes[:, 1]**2\n",
    "\n",
    "ax1.plot(tauval,data_l,c=\"blue\", linewidth=1.5, linestyle='-')\n",
    "ax2.plot(tauval,data_E,c=\"blue\", linewidth=1.5, linestyle='-')\n",
    "ax3.plot(tauval,data_ds2,c=\"blue\", linewidth=1.5, linestyle='-');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Klassifizierung unterschiedlicher Bahnbewegungen\n",
    "\n",
    "Literatur:\n",
    "\n",
    "<ol>\n",
    "   <li><i>General relativity : An introduction for physicists</i> von M. P. Hobson, G. P. Efstathiou und A. N. Lasenby</li>\n",
    "   <li><i>Gravity : An introduction to Einstein's general relativity</i> von James B. Hartle</li>\n",
    "   <li><i>Allgemeine Relativit&auml;tstheorie</i> von Torsten Flie&szlig;bach</li>\n",
    "   <li><i>Relativistic hydrodynamics</i> von Luciano Rezzolla und Olindo Zanotti</li>\n",
    "</ol>\n",
    "\n",
    "\n",
    "\n",
    "Man kann zeigen (siehe z.B. Seite 206 in <A href=\"https://doi.org/10.1017/CBO9780511790904\" target=\"_blank\">General relativity : An introduction for physicists by M. P. Hobson, G. P. Efstathiou and A. N. Lasenby</A>), dass sich die erste und vierte Gleichung des Systems der Differenzialgleichungen der Geodätengleichung in die folgenden Gleichungen umschreiben lässt:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\hbox{1. Gleichung:}&& \\frac{d}{d\\tau} \\left[  \\left( 1 - \\frac{2 M}{r} \\right) \\frac{dt}{d\\tau} \\right] = 0 \\\\\n",
    "\\rightarrow  && \\left( 1 - \\frac{2 M}{r}  \\right) \\frac{dt}{d\\tau} = e = \\hbox{const} \\\\\n",
    "\\hbox{4. Gleichung:}&& \\frac{d}{d\\tau}  \\left( r^2 \\hbox{sin}^2(\\theta)  \\frac{d\\phi}{d\\tau} \\right)   = 0 \\\\\n",
    "\\rightarrow &&  r^2 \\hbox{sin}^2(\\theta)  \\frac{d\\phi}{d\\tau}  = l = \\hbox{const} \\quad ,\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "wobei die w&auml;hrend der Bewegungen erhaltenen Gr&ouml;&szlig;en e (Teilchenenergie pro Masse, $e=\\frac{E}{m}$) und l (Drehimpuls pro Masse m, $l=\\frac{L}{m}$) sich aus der Definition des Viererimpulses $p_\\mu = m \\, u_\\mu = m \\, g_{\\mu\\nu} \\frac{dx^\\nu}{d\\tau} = m \\, \\left( g_{tt} \\, \\frac{dt}{d\\tau}, g_{rr} \\, \\frac{dr}{d\\tau}, g_{\\theta\\theta} \\, \\frac{d\\theta}{d\\tau}, g_{\\phi\\phi} \\, \\frac{d\\phi}{d\\tau}  \\right) $ ergeben (Voraussetzung: diagonale Form der Metrik). F&uuml;r die Schwarzschildmetrik erhält man:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "&& p_t = p_0 = m \\frac{dx_0}{d\\tau} = m g_{00} \\frac{dx^0}{d\\tau} = m g_{00} \\frac{dt}{d\\tau} = m \\left( 1 - \\frac{2 M}{r} \\right) \\frac{dt}{d\\tau} = m \\, e \\\\\n",
    "&& p_\\phi = p_3 = m \\frac{dx_3}{d\\tau} = m g_{33} \\frac{dx^3}{d\\tau} = m g_{33} \\frac{d\\phi}{d\\tau} = - m \\left( r^2 \\hbox{sin}^2(\\theta) \\right) \\frac{d\\phi}{d\\tau} = - m \\, l\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "\n",
    "Die Klassifikation m&ouml;glicher Bahnen von Probek&ouml;rpern in vorgegebener Schwarzschild-Raumzeit kann mittels eines definierten effektiven Potenzials illustriert werden. Dieses Potenzial h&auml;ngt von dem, bei der Bewegung erhaltenen Drehimpuls pro Masse m ab. Die im Zentralfeld m&ouml;glichen Bewegungen werden mittels zweier erhaltener Gr&ouml;&szlig;en (l: Drehimpuls pro Masse m und e: Energie pro Masse) charakterisiert. Die Definition des effektiven Potenzial erfolgt mittels der radialen, 2. Geod&auml;tengleichung. So definieren z.B. die Literaturangaben 1-3 (siehe Angaben oben) das effektive Potenzial wie folgt:\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\hbox{2. Gleichung:}&\\rightarrow&\\frac{1}{2} \\left( \\frac{dr}{d\\tau}  \\right)^2 + V(r,M,l) = \\frac{1}{2} \\left(  e^2 -1 \\right)  \\\\\n",
    "\\hbox{wobei:} && V(r,M,l) =   \\frac{l^2}{2 r^2} \\left(  1 - \\frac{2 M}{r} \\right) - \\frac{M}{r} \\quad .\n",
    "\\end{eqnarray}\n",
    "$$\n",
    "In der Literaturangabe 4 wird das Pozential V(r,M,l) hingegen wie folgt definiert:\n",
    "\n",
    "$$\n",
    "\\begin{eqnarray}\n",
    "\\hbox{2. Gleichung:}&\\rightarrow& \\left( \\frac{dr}{d\\tau}  \\right)^2 + \\left( V(r,M,l) \\right)^2 = e^2   \\\\\n",
    "\\hbox{wobei:} && V(r,M,l) =  \\sqrt{ \\left( 1 - \\frac{2 M}{r} \\right) \\left( 1 + \\frac{l^2}{r^2} \\right) } \\quad .\n",
    "\\end{eqnarray}\n",
    "$$  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = symbols('l')\n",
    "VeffFb = -M/r+l**2/(2*r**2)-M*l**2/r**3\n",
    "VeffRez = sqrt(1-2*M/r)*sqrt(1+l**2/r**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Drehimpuls l\n",
      "4.1000000000000005\n",
      "Energie e\n",
      "0.9666850572963256\n"
     ]
    }
   ],
   "source": [
    "setl = r0**2*dphi0\n",
    "setE = dt0*((1-2*set_M/r0))\n",
    "print(\"Drehimpuls l\")\n",
    "print(setl)\n",
    "print(\"Energie e\")\n",
    "print(setE)\n",
    "VeffFbl = lambdify((r),VeffFb.subs({(l,setl),(M,set_M)}))\n",
    "VeffRezl = lambdify((r),VeffRez.subs({(l,setl),(M,set_M)}))\n",
    "rval = np.linspace(2.85,30, 1001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",
      "text/plain": [
       "<Figure size 1600x800 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(16,8))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[1,1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0])\n",
    "ax2 = plt.subplot(gs[1])\n",
    "ax1.set_ylabel(r\"$\\rm V_1(r)$\")\n",
    "ax1.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "ax2.set_ylabel(r\"$\\rm V_2(r)$\")\n",
    "ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "ax1.plot(rval,VeffRezl(rval),c=\"blue\", linewidth=1.5, linestyle='-');\n",
    "ax2.plot(rval,VeffFbl(rval),c=\"red\", linewidth=1.5, linestyle='-');\n",
    "\n",
    "ax1.scatter(r0,VeffRezl(r0),c=\"black\", s=80, marker='o');\n",
    "ax2.scatter(r0,VeffFbl(r0),c=\"black\", s=80, marker='o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die obere Abbildung zeigt (in der Nomenklatur des 4.Buches (Diagramm links) bzw. des 1. bis 3. Buches (Diagramm rechts) ) das effektive Pozential als Funktion des Radius bei den zuvor gewählten Anfangswerten (schwarze Punkte auf den Kurven).\n",
    "\n",
    "Wir wollen uns nun die Bewegung des Probekörpers in einer Animation darstellen:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.animation as animation\n",
    "from IPython.display import HTML"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<video width=\"1100\" height=\"500\" controls autoplay loop>\n",
       "  <source type=\"video/mp4\" src=\"data:video/mp4;base64,AAAAIGZ0eXBNNFYgAAACAE00ViBpc29taXNvMmF2YzEAAAAIZnJlZQAA07ltZGF0AAACrwYF//+r\n",
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anzframes = 100\n",
    "step = int(N/anzframes-1) \n",
    "xylim = 18\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[1,1], wspace=0.3)\n",
    "ax1 = plt.subplot(gs[0])\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "def init():\n",
    "    ax1.scatter(Loes[0, 1]*np.cos(Loes[0, 2]),Loes[0, 1]*np.sin(Loes[0, 2]),c=\"red\", s=60, marker='o')\n",
    "    BH = plt.Circle((0, 0), 2, color='black')\n",
    "    ax1.add_patch(BH)\n",
    "    ax2.plot(rval,VeffRezl(rval),c=\"blue\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(r0,VeffRezl(r0),c=\"red\", s=80, marker='o')\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    ax1.cla() \n",
    "    ax2.cla() \n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),c=\"blue\", linewidth=0.5, linestyle='-')\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),c=\"red\", s=60, marker='o')\n",
    "    ax2.plot(rval,VeffRezl(rval),c=\"blue\", linewidth=1.5, linestyle='-')\n",
    "    ax2.scatter(Loes[step*i, 1],VeffRezl(r0),c=\"red\", s=80, marker='o')\n",
    "    BH = plt.Circle((0, 0), 2, color='black')\n",
    "    ax1.add_patch(BH);\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(fig,animate,init_func=init,frames=anzframes,interval=100)\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "bzw. ohne Verwendung von ax1.cla() und ax2.cla() in jedem Animationsschritt und mit einer Trennung von Animationsobjekten und statischen Elementen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Figure-Setup\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[1, 1], wspace=0.3)\n",
    "ax1 = plt.subplot(gs[0])\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "# Statische Elemente der Animation\n",
    "ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax1.set_xlim(-xylim, xylim)\n",
    "ax1.set_ylim(-xylim, xylim)\n",
    "ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "# Statische Potentialkurve\n",
    "ax2.plot(rval, VeffRezl(rval), c=\"blue\", linewidth=1.5, linestyle='-')\n",
    "# Statisches Schwarzes Loch\n",
    "BH = plt.Circle((0, 0), 2, color='black')\n",
    "ax1.add_patch(BH)\n",
    "\n",
    "# Animationsobjekte initialisieren\n",
    "line_orbit, = ax1.plot([], [], c=\"blue\", linewidth=0.5, linestyle='-')\n",
    "scatter_body = ax1.scatter([], [], c=\"red\", s=60, marker='o')\n",
    "scatter_pot = ax2.scatter([], [], c=\"red\", s=80, marker='o')\n",
    "\n",
    "# Simulationswerte der Trajektorie\n",
    "x_all = Loes[:, 1] * np.cos(Loes[:, 2])\n",
    "y_all = Loes[:, 1] * np.sin(Loes[:, 2])\n",
    "\n",
    "def init():\n",
    "    line_orbit.set_data([], [])\n",
    "    scatter_body.set_offsets(np.c_[x_all[0], y_all[0]])\n",
    "    scatter_pot.set_offsets(np.c_[Loes[0, 1], VeffRezl(Loes[0, 1])]) \n",
    "    return line_orbit, scatter_body, scatter_pot\n",
    "\n",
    "def animate(i):\n",
    "    idx = step * i\n",
    "    line_orbit.set_data(x_all[:idx], y_all[:idx])\n",
    "    scatter_body.set_offsets(np.c_[x_all[idx], y_all[idx]])\n",
    "    scatter_pot.set_offsets(np.c_[Loes[idx, 1], VeffRezl(Loes[0, 1])])\n",
    "    return line_orbit, scatter_body, scatter_pot\n",
    "\n",
    "ani = animation.FuncAnimation(\n",
    "    fig, animate, init_func=init, frames=anzframes, \n",
    "    interval=100, blit=True\n",
    ")\n",
    "\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir betrachten uns die gleiche Animation in einem eingebetteten Diagramm der räumlichen Hypersphäre der Mannigfaltigkeit:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib import cm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y = symbols('x,y')\n",
    "zBett = sqrt(8*M*(r-2*M))\n",
    "R = np.linspace(2.00001, 20, 300)\n",
    "PHI = np.linspace(0, 2*np.pi, 300)\n",
    "R, PHI = np.meshgrid(R, PHI)\n",
    "X = R*np.cos(PHI)\n",
    "Y = R*np.sin(PHI)\n",
    "zlam = lambdify((x,y), zBett.subs({(M,set_M),(r,sqrt(x**2+y**2))}))\n",
    "ZF = np.array(list(np.real(zlam(X,Y))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
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       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anzframes = 50\n",
    "step = int(N/anzframes-1) \n",
    "xylim = 18\n",
    "zlim = 15\n",
    "\n",
    "fig = plt.figure(figsize=(11,5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[2,1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0],projection='3d')\n",
    "ax2 = plt.subplot(gs[1])\n",
    "ax1.xaxis.pane.fill = False\n",
    "ax1.yaxis.pane.fill = False\n",
    "ax1.zaxis.pane.fill = False\n",
    "ax1.xaxis.pane.set_edgecolor('white')\n",
    "ax1.yaxis.pane.set_edgecolor('white')\n",
    "ax1.zaxis.pane.set_edgecolor('white')\n",
    "\n",
    "def init():\n",
    "    ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.5)\n",
    "    ax1.scatter(Loes[0, 1]*np.cos(Loes[0, 2]),Loes[0, 1]*np.sin(Loes[0, 2]),np.real(zlam(Loes[0, 1]*np.cos(Loes[0, 2]),Loes[0, 1]*np.sin(Loes[0, 2]))),c=\"black\", s=60, marker='o');\n",
    "    ax1.contourf(X, Y, ZF,100, zdir='z', offset=0, cmap=cm.nipy_spectral,alpha=0.5)\n",
    "    ax1.scatter(Loes[0, 1]*np.cos(Loes[0, 2]),Loes[0, 1]*np.sin(Loes[0, 2]),0.1,c=\"black\", s=40, marker='o');\n",
    "    ax2.plot(rval,VeffRezl(rval),c=\"black\", linewidth=1.5, linestyle='-');\n",
    "    ax2.scatter(r0,VeffRezl(r0),c=\"black\", s=80, marker='o');\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax1.set_zlabel(\"Visualisierung z\");\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.view_init(azim=35, elev=20)\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax1.set_zlim(0, 15)\n",
    "    ax1.grid(False)\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    ax1.cla() \n",
    "    ax2.cla()\n",
    "    ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.5)\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),np.real(zlam(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]))),c=\"black\", linewidth=1, linestyle='-');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),np.real(zlam(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]))),c=\"black\", s=60, marker='o');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),0.1,c=\"black\", s=40, marker='o');\n",
    "    ax1.contourf(X, Y, ZF,100, zdir='z', offset=0, cmap=cm.nipy_spectral,alpha=0.5)\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),0.1,c=\"black\", linewidth=2.5, linestyle='-');\n",
    "    ax2.plot(rval,VeffRezl(rval),c=\"black\", linewidth=1.5, linestyle='-');\n",
    "    ax2.scatter(Loes[step*i, 1],VeffRezl(r0),c=\"black\", s=80, marker='o');\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax1.set_zlabel(\"Einbettung z\");\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.view_init(azim=35, elev=20)\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax1.set_zlim(0, 15)  \n",
    "    ax1.grid(False)\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(fig,animate,init_func=init,frames=anzframes,interval=100)\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "bzw. ohne Verwendung von ax1.cla() und ax2.cla() in jedem Animationsschritt und mit einer Trennung von Animationsobjekten und statischen Elementen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "anzframes = 50\n",
    "step = int(N/anzframes-1) \n",
    "xylim = 18\n",
    "zlim = 15\n",
    "\n",
    "# Figure-Setup\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[2, 1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0], projection='3d')\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "for axis in [ax1.xaxis, ax1.yaxis, ax1.zaxis]:\n",
    "    axis.pane.fill = False\n",
    "    axis.pane.set_edgecolor('white')\n",
    "\n",
    "# Achsen-Konfiguration\n",
    "ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax1.set_zlabel(\"Einbettung z\")\n",
    "ax1.set_xlim(-xylim, xylim)\n",
    "ax1.set_ylim(-xylim, xylim)\n",
    "ax1.set_zlim(0, zlim)\n",
    "ax1.view_init(azim=35, elev=20)\n",
    "ax1.grid(False)\n",
    "ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "\n",
    "ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.5)\n",
    "ax1.contourf(X, Y, ZF, 100, zdir='z', offset=0, cmap=cm.nipy_spectral, alpha=0.5)\n",
    "ax2.plot(rval, VeffRezl(rval), c=\"black\", linewidth=1.5, linestyle='-')\n",
    "\n",
    "line_3d, = ax1.plot([], [], [], c=\"black\", linewidth=1, linestyle='-')       # Trajektorie auf Fläche\n",
    "line_shadow, = ax1.plot([], [], [], c=\"black\", linewidth=2.5, linestyle='-') # Schatten auf dem Boden\n",
    "\n",
    "scatter_3d = ax1.scatter([], [], [], c=\"black\", s=60, marker='o')\n",
    "scatter_shadow = ax1.scatter([], [], [], c=\"black\", s=40, marker='o')\n",
    "scatter_pot = ax2.scatter([], [], c=\"black\", s=80, marker='o')\n",
    "\n",
    "x_all = Loes[:, 1] * np.cos(Loes[:, 2])\n",
    "y_all = Loes[:, 1] * np.sin(Loes[:, 2])\n",
    "z_all = np.real(zlam(x_all, y_all))\n",
    "z_shadow = np.full_like(z_all, 0.1) \n",
    "\n",
    "def init():\n",
    "    line_3d.set_data([], [])\n",
    "    line_3d.set_3d_properties([])\n",
    "    line_shadow.set_data([], [])\n",
    "    line_shadow.set_3d_properties([])\n",
    "    \n",
    "    scatter_3d._offsets3d = ([], [], [])\n",
    "    scatter_shadow._offsets3d = ([], [], [])\n",
    "    scatter_pot.set_offsets(np.c_[[], []])\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    idx = step * i\n",
    "    line_3d.set_data(x_all[:idx], y_all[:idx])\n",
    "    line_3d.set_3d_properties(z_all[:idx])\n",
    "    line_shadow.set_data(x_all[:idx], y_all[:idx])\n",
    "    line_shadow.set_3d_properties(z_shadow[:idx])\n",
    "    \n",
    "    scatter_3d._offsets3d = ([x_all[idx]], [y_all[idx]], [z_all[idx]])\n",
    "    scatter_shadow._offsets3d = ([x_all[idx]], [y_all[idx]], [0.1])\n",
    "    scatter_pot.set_offsets(np.c_[Loes[idx, 1], VeffRezl(Loes[0, 1])])\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(\n",
    "    fig, animate, init_func=init, frames=anzframes, \n",
    "    interval=100, blit=True\n",
    ")\n",
    "\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Im Folgenden werden die unterschiedlichen Bahnbewegungen um ein schwarzes Loch klassifiziert. Die einzelnen Bewegungstypen lassen sich physikalisch über die Energie $e$ und den Drehimpuls $l$ des Testteilchens relativ zum effektiven Potenzial $V(r,M,l)$ wie folgt charakterisieren: "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A) <i>Kreisförmige Bahnen</i>: \n",
    "Diese Bahnen existieren an den Extrempunkten des effektiven Potenzials. Man unterscheidet hierbei zwischen stabilen kreisförmigen Bahnen (im Minimum des Potenzials, oberhalb des sogenannten ISCO – Innermost Stable Circular Orbit) und instabilen kreisförmigen Bahnen (auf dem Maximum des Potentials)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "t0 = 0\n",
    "phi0 = 0\n",
    "dr0 = 0\n",
    "setl = 4.1\n",
    "setVeffRez = VeffRez.subs({(l,setl),(M,set_M)})\n",
    "EqCircOrb = Eq(diff(setVeffRez,r),0)\n",
    "r0A = float(solve(EqCircOrb)[1])\n",
    "dphi0A = setl/r0A**2\n",
    "dt0A = float(LoesEq_ds[0].subs({(r,r0A),(dr,dr0),(dphi,dphi0A),(M,set_M)}))\n",
    "setEA = dt0A*((1-2*set_M/r0A))\n",
    "initialvalA = np.array([t0,r0A,phi0,dt0A,dr0,dphi0A])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "B) <i>Elliptische Bahnen / Rosettenbahnen</i>: \n",
    "Wenn die Energie des Probek&ouml;rpers größer als das Minimum, aber kleiner als das Maximum des Potenzials ist, resultiert eine gebundene Bewegung. Aufgrund der relativistischen Effekte schließt sich diese Bahn jedoch nicht wie bei Newton (Kepler-Ellipse), sondern führt zu einer markanten Periheldrehung (Einstein-Präzession), wodurch die typische Rosettenform entsteht."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "r0B = 8\n",
    "dphi0B = setl/r0B**2\n",
    "dt0B = float(LoesEq_ds[0].subs({(r,r0B),(dr,dr0),(dphi,dphi0B),(M,set_M)}))\n",
    "setEB = dt0B*(1-2*set_M/r0B)\n",
    "initialvalB = np.array([t0,r0B,phi0,dt0B,dr0,dphi0B])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "C) <i>Ungebunden parabolische Bahnen</i>: \n",
    "Diese stellen den Grenzfall dar, bei dem der Probek&ouml;rper exakt die Fluchtenergie besitzt: $e = \\lim\\limits_{r \\to \\infty}V(r,M,l) = 1$ . Das Teilchen kommt theoretisch aus dem Unendlichen mit der Geschwindigkeit Null an, wird vom Schwarzen Loch abgelenkt und entweicht wieder ins Unendliche."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "setEC = float(limit_seq(setVeffRez, r))\n",
    "EqE1 = Eq(setVeffRez,setEC)\n",
    "r0C = float(solve(EqE1,r)[1])\n",
    "dphi0C = setl/r0C**2\n",
    "dt0C = float(LoesEq_ds[0].subs({(r,r0C),(dr,dr0),(dphi,dphi0C),(M,set_M)}))\n",
    "initialvalC = np.array([t0,r0C,phi0,dt0C,dr0,dphi0C])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "D) <i>Ungebunden hyperbolische Bahnen</i>:\n",
    "Hier besitzt das Teilchen eine Überschussenergie: $e > \\lim\\limits_{r \\to \\infty}V(r,M,l) = 1$. Es nähert sich mit einer hohen Anfangsgeschwindigkeit aus dem Unendlichen, erfährt in der Nähe des Ereignishorizonts eine starke gravitative Streuung (oft mit extremen relativistischen Ablenkungswinkeln) und verlässt das System wieder."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "setED = 1.005\n",
    "EqE1005 = Eq(setVeffRez,setED)\n",
    "r0D = solve(EqE1005,r)[2]\n",
    "r0D = float(re(r0D))\n",
    "dphi0D = setl/r0D**2\n",
    "dt0D = float(LoesEq_ds[0].subs({(r,r0D),(dr,dr0),(dphi,dphi0D),(M,set_M)}))\n",
    "initialvalD = np.array([t0,r0D,phi0,dt0D,dr0,dphi0D])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 4.75972584736247$"
      ],
      "text/plain": [
       "4.759725847362467"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r0D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "E) <i>Einfangbahnen / Capture Orbits</i>:\n",
    "Wenn die Energie oder der Drehimpuls des Teilchens so beschaffen sind, dass es die Potenzialbarriere des Schwarzen Lochs überwindet (Energie $e$ ist größer als das Maximum des Potentials), gibt es keinen klassischen Umkehrpunkt mehr. Das Teilchen spiralisiert unaufhaltsam über den Ereignishorizont hinaus und wird vom Schwarzen Loch dauerhaft verschluckt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "r0E = 225\n",
    "dphi0E = setl/r0E**2\n",
    "dr0E=-0.185388257\n",
    "dt0E = float(LoesEq_ds[0].subs({(r,r0E),(dr,dr0E),(dphi,dphi0E),(M,set_M)}))\n",
    "setEE = dt0E*((1-2*set_M/r0E))\n",
    "initialvalE = np.array([t0,r0E,phi0,dt0E,dr0E,dphi0E])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Wir simulieren alle fünf Bahnbewegungen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 50000\n",
    "tauval = np.linspace(0, 1068, N+1)\n",
    "LoesA = integrate.odeint(DGLsys, initialvalA, tauval)\n",
    "LoesB = integrate.odeint(DGLsys, initialvalB, tauval)\n",
    "LoesC = integrate.odeint(DGLsys, initialvalC, tauval)\n",
    "LoesD = integrate.odeint(DGLsys, initialvalD, tauval)\n",
    "LoesE = integrate.odeint(DGLsys, initialvalE, tauval)\n",
    "Loes_vec = [LoesA,LoesB,LoesC,LoesD,LoesE]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "zBett = sqrt(8*M*(r-2*M))\n",
    "R = np.linspace(2.00001, 40, 1000)\n",
    "PHI = np.linspace(0, 2*np.pi, 300)\n",
    "R, PHI = np.meshgrid(R, PHI)\n",
    "X = R*np.cos(PHI)\n",
    "Y = R*np.sin(PHI)\n",
    "zlam = lambdify((x,y), zBett.subs({(M,1),(r,sqrt(x**2+y**2))}))\n",
    "ZF = np.array(list(np.real(zlam(X,Y))))\n",
    "rval = np.linspace(2.85,40, 1001)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step = 500\n",
    "anzframes = 100\n",
    "xylim = 35\n",
    "zlim = 20\n",
    "fig = plt.figure(figsize=(11,5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[2,1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0],projection='3d')\n",
    "ax2 = plt.subplot(gs[1])\n",
    "ax1.xaxis.pane.fill = False\n",
    "ax1.yaxis.pane.fill = False\n",
    "ax1.zaxis.pane.fill = False\n",
    "ax1.xaxis.pane.set_edgecolor('white')\n",
    "ax1.yaxis.pane.set_edgecolor('white')\n",
    "ax1.zaxis.pane.set_edgecolor('white')\n",
    "\n",
    "def init():\n",
    "    ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.5)    \n",
    "    ax1.contourf(X, Y, ZF,100, zdir='z', offset=0, cmap=cm.nipy_spectral,alpha=0.5)\n",
    "    ax1.scatter(LoesA[0, 1]*np.cos(LoesA[0, 2]),LoesA[0, 1]*np.sin(LoesA[0, 2]),np.real(zlam(LoesA[0, 1]*np.cos(LoesA[0, 2]),LoesA[0, 1]*np.sin(LoesA[0, 2]))),c=\"blue\", s=60, marker='o');\n",
    "    ax1.scatter(LoesA[0, 1]*np.cos(LoesA[0, 2]),LoesA[0, 1]*np.sin(LoesA[0, 2]),0.1,c=\"blue\", s=40, marker='o');\n",
    "    ax1.scatter(LoesB[0, 1]*np.cos(LoesB[0, 2]),LoesB[0, 1]*np.sin(LoesB[0, 2]),np.real(zlam(LoesB[0, 1]*np.cos(LoesB[0, 2]),LoesB[0, 1]*np.sin(LoesB[0, 2]))),c=\"red\", s=60, marker='o');\n",
    "    ax1.scatter(LoesB[0, 1]*np.cos(LoesB[0, 2]),LoesB[0, 1]*np.sin(LoesB[0, 2]),0.1,c=\"red\", s=40, marker='o');\n",
    "    ax1.scatter(LoesC[0, 1]*np.cos(LoesC[0, 2]),LoesC[0, 1]*np.sin(LoesC[0, 2]),np.real(zlam(LoesC[0, 1]*np.cos(LoesC[0, 2]),LoesC[0, 1]*np.sin(LoesC[0, 2]))),c=\"green\", s=60, marker='o');\n",
    "    ax1.scatter(LoesC[0, 1]*np.cos(LoesC[0, 2]),LoesC[0, 1]*np.sin(LoesC[0, 2]),0.1,c=\"green\", s=40, marker='o');\n",
    "    ax1.scatter(LoesD[0, 1]*np.cos(LoesD[0, 2]),LoesD[0, 1]*np.sin(LoesD[0, 2]),np.real(zlam(LoesD[0, 1]*np.cos(LoesD[0, 2]),LoesD[0, 1]*np.sin(LoesD[0, 2]))),c=\"grey\", s=60, marker='o');\n",
    "    ax1.scatter(LoesD[0, 1]*np.cos(LoesD[0, 2]),LoesD[0, 1]*np.sin(LoesD[0, 2]),0.1,c=\"grey\", s=40, marker='o');\n",
    "    ax1.scatter(LoesE[0, 1]*np.cos(LoesE[0, 2]),LoesE[0, 1]*np.sin(LoesE[0, 2]),np.real(zlam(LoesE[0, 1]*np.cos(LoesE[0, 2]),LoesE[0, 1]*np.sin(LoesE[0, 2]))),c=\"black\", s=60, marker='o');\n",
    "    ax1.scatter(LoesE[0, 1]*np.cos(LoesE[0, 2]),LoesE[0, 1]*np.sin(LoesE[0, 2]),0.1,c=\"black\", s=40, marker='o');\n",
    "    ax2.plot(rval,VeffRezl(rval),c=\"black\", linewidth=1.5, linestyle='-');\n",
    "    ax2.scatter(r0,VeffRezl(r0),c=\"black\", s=80, marker='o');\n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax1.set_zlabel(\"Visualisierung z\");\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.view_init(azim=-130, elev=35)\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax1.set_zlim(0, zlim)\n",
    "    ax2.set_xlim(2, xylim)\n",
    "    ax1.grid(False)\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    ax1.cla() \n",
    "    ax2.cla()\n",
    "    ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.5)\n",
    "    ax1.contourf(X, Y, ZF,100, zdir='z', offset=0, cmap=cm.nipy_spectral,alpha=0.5)\n",
    "    Loes = LoesA\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),np.real(zlam(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]))),c=\"blue\", linewidth=1, linestyle='-');\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),0.1,c=\"blue\", linewidth=2.5, linestyle='-');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),np.real(zlam(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]))),c=\"blue\", s=60, marker='o');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),0.1,c=\"blue\", s=40, marker='o');\n",
    "    Loes = LoesB\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),np.real(zlam(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]))),c=\"red\", linewidth=1, linestyle='-');\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),0.1,c=\"red\", linewidth=2.5, linestyle='-');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),np.real(zlam(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]))),c=\"red\", s=60, marker='o');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),0.1,c=\"red\", s=40, marker='o');\n",
    "    Loes = LoesC\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),np.real(zlam(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]))),c=\"green\", linewidth=1, linestyle='-');\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),0.1,c=\"green\", linewidth=2.5, linestyle='-');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),np.real(zlam(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]))),c=\"green\", s=60, marker='o');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),0.1,c=\"green\", s=40, marker='o');\n",
    "    Loes = LoesD\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),np.real(zlam(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]))),c=\"grey\", linewidth=1, linestyle='-');\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),0.1,c=\"grey\", linewidth=2.5, linestyle='-');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),np.real(zlam(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]))),c=\"grey\", s=60, marker='o');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),0.1,c=\"grey\", s=40, marker='o');\n",
    "    Loes = LoesE\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),np.real(zlam(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]))),c=\"black\", linewidth=1, linestyle='-');\n",
    "    ax1.plot(Loes[:step*i, 1]*np.cos(Loes[:step*i, 2]),Loes[:step*i, 1]*np.sin(Loes[:step*i, 2]),0.1,c=\"black\", linewidth=2.5, linestyle='-');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),np.real(zlam(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]))),c=\"black\", s=60, marker='o');\n",
    "    ax1.scatter(Loes[step*i, 1]*np.cos(Loes[step*i, 2]),Loes[step*i, 1]*np.sin(Loes[step*i, 2]),0.1,c=\"black\", s=40, marker='o');\n",
    "\n",
    "    ax2.plot(rval,VeffRezl(rval),c=\"black\", linewidth=1.5, linestyle='-');\n",
    "    ax2.scatter(LoesA[step*i, 1],VeffRezl(LoesA[0, 1]),c=\"blue\", s=80, marker='o');\n",
    "    ax2.scatter(LoesB[step*i, 1],VeffRezl(LoesB[0, 1]),c=\"red\", s=80, marker='o');\n",
    "    ax2.plot(LoesB[:step*i, 1],VeffRezl(LoesB[0, 1]*LoesB[:step*i, 1]/LoesB[:step*i, 1]),c=\"red\", linewidth=1, linestyle=':');\n",
    "    ax2.scatter(LoesC[step*i, 1],VeffRezl(LoesC[0, 1]),c=\"green\", s=80, marker='o');\n",
    "    ax2.plot(LoesC[:step*i, 1],VeffRezl(LoesC[0, 1]*LoesC[:step*i, 1]/LoesC[:step*i, 1]),c=\"green\", linewidth=1, linestyle=':');\n",
    "    ax2.scatter(LoesD[step*i, 1],VeffRezl(LoesD[0, 1]),c=\"grey\", s=80, marker='o');\n",
    "    ax2.plot(LoesD[:step*i, 1],VeffRezl(LoesD[0, 1]*LoesD[:step*i, 1]/LoesD[:step*i, 1]),c=\"grey\", linewidth=1, linestyle=':');\n",
    "    ax2.scatter(LoesE[step*i, 1],setEE,c=\"black\", s=80, marker='o');\n",
    "    ax2.plot(LoesE[:step*i, 1],setEE*LoesE[:step*i, 1]/LoesE[:step*i, 1],c=\"black\", linewidth=1, linestyle=':');\n",
    "    \n",
    "    \n",
    "    ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "    ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "    ax1.set_zlabel(\"Einbettung z\");\n",
    "    ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "    ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "    ax1.view_init(azim=-130, elev=35)\n",
    "    ax1.set_xlim(-xylim, xylim)\n",
    "    ax1.set_ylim(-xylim, xylim)\n",
    "    ax1.set_zlim(0, zlim)  \n",
    "    ax2.set_xlim(2, xylim)\n",
    "    ax1.grid(False)\n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(fig,animate,init_func=init,frames=anzframes,interval=100)\n",
    "plt.close(ani._fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "bzw. ohne Verwendung von ax1.cla() und ax2.cla() in jedem Animationsschritt und mit einer Trennung von Animationsobjekten und statischen Elementen."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "YXZmNjAuMTYuMTAw\n",
       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "step = 500\n",
    "anzframes = 100\n",
    "xylim = 35\n",
    "zlim = 20\n",
    "\n",
    "# Figure-Setup\n",
    "fig = plt.figure(figsize=(11, 5))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[2, 1], wspace=0.21)\n",
    "ax1 = plt.subplot(gs[0], projection='3d')\n",
    "ax2 = plt.subplot(gs[1])\n",
    "\n",
    "for axis in [ax1.xaxis, ax1.yaxis, ax1.zaxis]:\n",
    "    axis.pane.fill = False\n",
    "    axis.pane.set_edgecolor('white')\n",
    "\n",
    "ax1.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax1.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax1.set_zlabel(\"Einbettung z\")\n",
    "ax1.set_xlim(-xylim, xylim)\n",
    "ax1.set_ylim(-xylim, xylim)\n",
    "ax1.set_zlim(0, zlim)\n",
    "ax1.view_init(azim=-130, elev=35)\n",
    "ax1.grid(False)\n",
    "\n",
    "v_values = VeffRezl(rval)\n",
    "ax2.set_ylabel(r\"$\\rm V(r)$\")\n",
    "ax2.set_xlabel(r\"$\\rm Radius \\, r \\,[km]$\")\n",
    "ax2.set_xlim(np.min(rval), np.max(rval))\n",
    "\n",
    "# Statische Hintergründe zeichnen\n",
    "ax1.plot_surface(X, Y, ZF, cmap=cm.nipy_spectral, linewidth=0, alpha=0.5)\n",
    "ax1.contourf(X, Y, ZF, 100, zdir='z', offset=0, cmap=cm.nipy_spectral, alpha=0.5)\n",
    "ax2.plot(rval, v_values, c=\"black\", linewidth=1.5, linestyle='-')\n",
    "\n",
    "# Animationsobjekte\n",
    "lines_3d = []\n",
    "lines_shadow = []\n",
    "scatters_3d = []\n",
    "scatters_shadow = []\n",
    "scatters_pot = []\n",
    "lines_pot = []\n",
    "\n",
    "colors = [\"blue\", \"red\", \"green\", \"gray\", \"black\"]\n",
    "for c in colors:\n",
    "    l3d, = ax1.plot([], [], [], c=c, linewidth=1.2, linestyle='-')\n",
    "    lsh, = ax1.plot([], [], [], c=c, linewidth=2.0, linestyle='-', alpha=0.6)\n",
    "    lines_3d.append(l3d)\n",
    "    lines_shadow.append(lsh)\n",
    "    scatters_3d.append(ax1.scatter([], [], [], c=c, s=50, marker='o'))\n",
    "    scatters_shadow.append(ax1.scatter([], [], [], c=c, s=30, marker='o'))\n",
    "    scatters_pot.append(ax2.scatter([], [], c=c, s=60, marker='o', zorder=5))\n",
    "    lpot, = ax2.plot([], [], c=c, linewidth=1, linestyle=':', alpha=0.7)\n",
    "    lines_pot.append(lpot)\n",
    "\n",
    "x_all_vec = []\n",
    "y_all_vec = []\n",
    "z_all_vec = []\n",
    "for Loes in Loes_vec:\n",
    "    x_all = Loes[:, 1] * np.cos(Loes[:, 2])\n",
    "    y_all = Loes[:, 1] * np.sin(Loes[:, 2])\n",
    "    z_all = np.real(zlam(x_all, y_all))   \n",
    "    x_all_vec.append(x_all)\n",
    "    y_all_vec.append(y_all)\n",
    "    z_all_vec.append(z_all)\n",
    "\n",
    "def init():\n",
    "    for j in range(5):\n",
    "        lines_3d[j].set_data([], [])\n",
    "        lines_3d[j].set_3d_properties([])\n",
    "        lines_shadow[j].set_data([], [])\n",
    "        lines_shadow[j].set_3d_properties([])  \n",
    "        scatters_3d[j]._offsets3d = ([], [], [])\n",
    "        scatters_shadow[j]._offsets3d = ([], [], [])\n",
    "        scatters_pot[j].set_offsets(np.empty((0, 2)))\n",
    "        lines_pot[j].set_data([], [])\n",
    "    return fig,\n",
    "\n",
    "def animate(i):\n",
    "    idx = step * i\n",
    "    for j in range(5):\n",
    "        Loes = Loes_vec[j]\n",
    "        x_all = x_all_vec[j]\n",
    "        y_all = y_all_vec[j]\n",
    "        z_all = z_all_vec[j]\n",
    "        \n",
    "        lines_3d[j].set_data(x_all[:idx], y_all[:idx])\n",
    "        lines_3d[j].set_3d_properties(z_all[:idx])\n",
    "        lines_shadow[j].set_data(x_all[:idx], y_all[:idx])\n",
    "        lines_shadow[j].set_3d_properties(np.full_like(x_all[:idx], 0.1))\n",
    "        scatters_3d[j]._offsets3d = ([x_all[idx]], [y_all[idx]], [z_all[idx]])\n",
    "        scatters_shadow[j]._offsets3d = ([x_all[idx]], [y_all[idx]], [0.1])\n",
    "        \n",
    "        r_hist = Loes[:idx, 1]\n",
    "        if(j < 4):\n",
    "            v_hist = VeffRezl(Loes[0, 1])\n",
    "        else:\n",
    "            v_hist = setEE  \n",
    "        if(len(r_hist) > 0):\n",
    "            scatters_pot[j].set_offsets(np.c_[r_hist[-1], v_hist])\n",
    "            lines_pot[j].set_data(r_hist, [v_hist]*len(r_hist))    \n",
    "    return fig,\n",
    "\n",
    "ani = animation.FuncAnimation(\n",
    "    fig, animate, init_func=init, frames=anzframes, \n",
    "    interval=100, blit=True\n",
    ")\n",
    "\n",
    "plt.close(fig)\n",
    "HTML(ani.to_html5_video())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In der oberen Animation erkennt man neben den gebundenen kreisförmigen (A: blau) und elliptischen (B: rot) Bahnen, den parabolischen (C: grün) und hyperbolischen (D: grau) Bahnverläufen auch eine durch das schwarze Loch eingefangene Bahn (E: capture orbit, schwarz). Diese ist in der unteren Abbildung nochmals separat veranschaulicht."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Loes = LoesE\n",
    "ax = plt.gca()\n",
    "ax.cla() \n",
    "ax.set_ylabel(r\"$\\rm y \\,[km]$\")\n",
    "ax.set_xlabel(r\"$\\rm x \\,[km]$\")\n",
    "ax.plot(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]),c=\"black\", linewidth=1.5, linestyle='-');\n",
    "ax.scatter(Loes[0, 1]*np.cos(Loes[0, 2]),Loes[0, 1]*np.sin(Loes[0, 2]),c=\"black\", s=60, marker='o');\n",
    "ax.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Die Visualisierung kann man auch mittels der \"Plotly Python Open Source Graphing Library\" erstellen ( siehe <A href=\"https://plotly.com/python/\" target=\"_blank\">https://plotly.com/python/</A> ). In der folgenden interaktiven Abbildung wird der \"Capture orbit\" und der \"Elliptische Orbit\" visualisiert.  Zusätzlich wird die Eigen- und Koordinatenzeit ausgegeben, wenn man mit der Maus genau auf eine der beiden Probekörper-Trajektorien fokussiert. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "import plotly.graph_objects as go"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<iframe\n",
       "    scrolling=\"no\"\n",
       "    width=\"720px\"\n",
       "    height=\"720\"\n",
       "    src=\"iframe_figures/figure_57.html\"\n",
       "    frameborder=\"0\"\n",
       "    allowfullscreen\n",
       "></iframe>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "trichter = go.Surface(x=X, y=Y, z=ZF, colorscale=\"spectral_r\", showscale=True, name=\"Embedded Surface\", hoverinfo='none')\n",
    "\n",
    "Loes = LoesE\n",
    "labels = []\n",
    "for i in range(len(Loes[:, 0])):\n",
    "    labels.append('Eigenzeit '+str(round(tauval[i], 2))+', Koordinatenzeit = '+str(round(Loes[i, 0], 2)))\n",
    "capture_orbit =go.Scatter3d(\n",
    "    x = Loes[:, 1]*np.cos(Loes[:, 2]),\n",
    "    y = Loes[:, 1]*np.sin(Loes[:, 2]),\n",
    "    z = np.real(zlam(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]))),\n",
    "    name = \"Capture Orbit\",\n",
    "    mode = 'lines',\n",
    "    line = dict(color='black', width=8),\n",
    "    text = labels,\n",
    "    hoverinfo = 'text'\n",
    "    )\n",
    "\n",
    "Loes = LoesB\n",
    "labels = []\n",
    "for i in range(len(Loes[:, 0])):\n",
    "    labels.append('Eigenzeit '+str(round(tauval[i], 2))+', Koordinatenzeit = '+str(round(Loes[i, 0], 2)))\n",
    "elliptic_orbit =go.Scatter3d(\n",
    "    x = Loes[:, 1]*np.cos(Loes[:, 2]),\n",
    "    y = Loes[:, 1]*np.sin(Loes[:, 2]),\n",
    "    z = np.real(zlam(Loes[:, 1]*np.cos(Loes[:, 2]),Loes[:, 1]*np.sin(Loes[:, 2]))),\n",
    "    name = \"Elliptische Bahn\",\n",
    "    mode = 'lines',\n",
    "    line = dict(color='red', width=8),\n",
    "    text = labels,\n",
    "    hoverinfo = 'text'\n",
    "    )\n",
    "\n",
    "layout = go.Layout(autosize=True,\n",
    "                  width=700, height=700,\n",
    "                  margin=dict(l=15, r=50, b=65, t=90),\n",
    "                  scene_aspectmode='cube',\n",
    "                  scene = dict(\n",
    "                    xaxis_showbackground = False,\n",
    "                    yaxis_showbackground = False,\n",
    "                    zaxis_showbackground = False,\n",
    "                    xaxis_gridcolor = \"lightgrey\",\n",
    "                    yaxis_gridcolor = \"lightgrey\",\n",
    "                    zaxis_gridcolor = \"lightgrey\",\n",
    "                    xaxis_range = [-40,40],\n",
    "                    yaxis_range = [-40,40],\n",
    "                    zaxis_range = [0,20],\n",
    "                    xaxis_title = 'x',\n",
    "                    yaxis_title = 'y',\n",
    "                    zaxis_title = 'f'),\n",
    "                  legend=dict(\n",
    "                    yanchor = \"top\",\n",
    "                    y=0.99,\n",
    "                    xanchor = \"left\",\n",
    "                    x=0.01)\n",
    "            )\n",
    "\n",
    "data = [trichter,elliptic_orbit,capture_orbit]\n",
    "fig = go.Figure(data = data, layout = layout)\n",
    "\n",
    "fig.show(renderer='iframe')"
   ]
  }
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