orbital-mechanics.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "a263016a-cdb7-44ca-83d2-29c46fa7a11b",
   "metadata": {},
   "source": [
    "# Simulating Orbital Mechanics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "id": "e0d59bec-44fc-48cd-8abb-116c52aaf041",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import sympy as sp\n",
    "import sympy.physics.mechanics as spm\n",
    "import scipy.integrate as spi\n",
    "import matplotlib.pyplot as plt\n",
    "sp.init_printing()\n",
    "spm.init_vprinting()\n",
    "dispForm = lambda lab, form: display(sp.Eq(sp.Symbol(lab), form))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8ec73c44-df45-42ab-bcf2-88dfaea2b9d2",
   "metadata": {},
   "source": [
    "## 2D Two Body Problem\n",
    "\n",
    "![Two Body Problem](resources/two-body-orbital-diagram.svg)\n",
    "\n",
    "System has rotational symmetry. Using polar coordinates $r_1\\ \\theta_1\\ r_2\\ \\theta_2$ and derivatives $\\dot{r}_1\\ \\dot{\\theta}_1\\ \\dot{r}_2\\ \\dot{\\theta}_2$. We can declare $s$ as the distance between both planets. $s$ is described by the following formula:\n",
    "\n",
    "$$\n",
    "s = \\sqrt{ r_1^2 + r_2^2 - 2 r_1 r_2 \\cos{ \\left( \\theta_2 - \\theta_1 \\right) } }\n",
    "$$\n",
    "\n",
    "We'll keep $s$ as is in order to make the derivations simpler and more elegant."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "id": "efd5cd76-044c-4938-9625-b6315c83891e",
   "metadata": {},
   "outputs": [],
   "source": [
    "G, m1, m2 = sp.symbols('G m_1 m_2')\n",
    "t = sp.symbols('t')\n",
    "r1, r2, th1, th2 = spm.dynamicsymbols('r_1 r_2 \\\\theta_1 \\\\theta_2')\n",
    "r1d, r2d, th1d, th2d = tuple( x.diff() for x in (r1, r2, th1, th2) )\n",
    "s = sp.Function('s')(r1, r2, th1, th2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11dfe9cd-6e40-4b33-b2b1-bdb3a58473ad",
   "metadata": {},
   "source": [
    "Since we're using Lagrangian mechanics, we derive the Lagrangian based on kinetic and potential energy. Potential energy relies only on the $s$ parameter, which we will try to keep as is in the final equations of motion so we won't have to compute them over and over again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "id": "86b53d91-7035-4c00-8152-cd70f2e74b77",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle L = \\frac{G m_{1} m_{2}}{s{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} + \\frac{m_{1} \\left(r_{1}^{2} \\dot{\\theta}_{1}^{2} + \\dot{r}_{1}^{2}\\right)}{2} + \\frac{m_{2} \\left(r_{2}^{2} \\dot{\\theta}_{2}^{2} + \\dot{r}_{2}^{2}\\right)}{2}$"
      ],
      "text/plain": [
       "                                     ⎛  2        2     2⎞      ⎛  2        2  \n",
       "              G⋅m₁⋅m₂             m₁⋅⎝r₁ ⋅\\thėta₁  + r₁̇ ⎠   m₂⋅⎝r₂ ⋅\\thėta₂\n",
       "L = ─────────────────────────── + ─────────────────────── + ──────────────────\n",
       "    s(r₁, r₂, \\theta₁, \\theta₂)              2                         2      \n",
       "\n",
       "   2⎞\n",
       "  + r₂̇ ⎠\n",
       "─────\n",
       "     "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "L = (m1 * (r1d**2 + (r1*th1d)**2) + m2 * (r2d**2 + (r2*th2d)**2)) / 2 + G * m1 * m2 / s\n",
    "dispForm('L', L)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "208dd4c8-9f68-4662-af34-4b9948dfe150",
   "metadata": {},
   "source": [
    "We plug this formula into the Euler-Lagrange equation to get our output solutions for each degree of freedom\n",
    "\n",
    "$$\n",
    "\\frac{\\partial L}{\\partial q_i} = \\frac{d}{dt} \\left( \\frac{\\partial L}{\\partial \\dot{q}_i} \\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "id": "08172f66-bbb2-4093-afd7-e92885ced580",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{r_1} = - \\frac{G m_{2} \\frac{d}{d r_{1}} s{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}}{s^{2}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} + r_{1} \\dot{\\theta}_{1}^{2}$"
      ],
      "text/plain": [
       "                     d                                            \n",
       "               G⋅m₂⋅───(s(r₁, r₂, \\theta₁, \\theta₂))              \n",
       "                    dr₁                                          2\n",
       "\\ddot{r_1} = - ───────────────────────────────────── + r₁⋅\\thėta₁ \n",
       "                     2                                            \n",
       "                    s (r₁, r₂, \\theta₁, \\theta₂)                  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{r_2} = - \\frac{G m_{1} \\frac{d}{d r_{2}} s{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}}{s^{2}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} + r_{2} \\dot{\\theta}_{2}^{2}$"
      ],
      "text/plain": [
       "                     d                                            \n",
       "               G⋅m₁⋅───(s(r₁, r₂, \\theta₁, \\theta₂))              \n",
       "                    dr₂                                          2\n",
       "\\ddot{r_2} = - ───────────────────────────────────── + r₂⋅\\thėta₂ \n",
       "                     2                                            \n",
       "                    s (r₁, r₂, \\theta₁, \\theta₂)                  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{\\theta_1} = - \\frac{G m_{2} \\frac{d}{d \\theta_{1}} s{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}}{r_{1}^{2} s^{2}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} - \\frac{2 \\dot{\\theta}_{1} \\dot{r}_{1}}{r_{1}}$"
      ],
      "text/plain": [
       "                            d                                                \n",
       "                    G⋅m₂⋅────────(s(r₁, r₂, \\theta₁, \\theta₂))               \n",
       "                         d\\theta₁                                2⋅\\thėta₁⋅r₁̇\n",
       "\\ddot{\\theta_1} = - ────────────────────────────────────────── - ────────────\n",
       "                           2  2                                       r₁     \n",
       "                         r₁ ⋅s (r₁, r₂, \\theta₁, \\theta₂)                    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{\\theta_2} = - \\frac{G m_{1} \\frac{d}{d \\theta_{2}} s{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}}{r_{2}^{2} s^{2}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} - \\frac{2 \\dot{\\theta}_{2} \\dot{r}_{2}}{r_{2}}$"
      ],
      "text/plain": [
       "                            d                                                \n",
       "                    G⋅m₁⋅────────(s(r₁, r₂, \\theta₁, \\theta₂))               \n",
       "                         d\\theta₂                                2⋅\\thėta₂⋅r₂̇\n",
       "\\ddot{\\theta_2} = - ────────────────────────────────────────── - ────────────\n",
       "                           2  2                                       r₂     \n",
       "                         r₂ ⋅s (r₁, r₂, \\theta₁, \\theta₂)                    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def solve_euler_lagrange(q, qd):\n",
    "    \"\"\"\n",
    "    Return solution to Euler-Lagrange equation \n",
    "    for lagrangian wrt coordinate and respective \n",
    "    time derivative\n",
    "    \"\"\"\n",
    "    eL = L.diff(q) - L.diff(qd).diff(t)\n",
    "    return sp.solve(eL, qd.diff())[0]\n",
    "\n",
    "r1dd  = solve_euler_lagrange(r1, r1d)\n",
    "r2dd  = solve_euler_lagrange(r2, r2d)\n",
    "th1dd = solve_euler_lagrange(th1, th1d)\n",
    "th2dd = solve_euler_lagrange(th2, th2d)\n",
    "dispForm('\\\\ddot{r_1}',       r1dd)\n",
    "dispForm('\\\\ddot{r_2}',       r2dd)\n",
    "dispForm('\\\\ddot{\\\\theta_1}', th1dd)\n",
    "dispForm('\\\\ddot{\\\\theta_2}', th2dd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f15ef49-71d0-420f-8587-8fd5bdd78990",
   "metadata": {},
   "source": [
    "We'll take a detour to solve for those derivatives of $s$. Using the chain rule, we can unravel this derivative piece by piece, keeping the $s$ term in wherever we can. We'll start with the square root term\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\frac{\\partial s}{\\partial q} &= \\frac{\\partial}{\\partial q} \\sqrt{ r_1^2 + r_2^2 - 2 r_1 r_2 \\cos{ \\left( \\theta_2 - \\theta_1 \\right) } } \\\\\n",
    "&= \\frac{\\frac{\\partial}{\\partial q} \\left( r_1^2 + r_2^2 - 2 r_1 r_2 \\cos{ \\left( \\theta_2 - \\theta_1 \\right) } \\right)}{2 \\sqrt{ r_1^2 + r_2^2 - 2 r_1 r_2 \\cos{ \\left( \\theta_2 - \\theta_1 \\right) } } } \\\\\n",
    "&= \\frac{ \\frac{\\partial}{\\partial q} \\left( r_1^2 + r_2^2 - 2 r_1 r_2 \\cos{ \\left( \\theta_2 - \\theta_1 \\right) } \\right) }{2s}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "From this, we can eliminate the square root term from the derivative, and, in fact, replace it with our $s$ term. The rest of the derivative depends on what's inside the square root. The derivatives for each of the coordinates are as follows.\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\frac{\\partial s}{\\partial r_1} &= \\frac{r_1 - r_2\\cos{\\left( \\theta_2 - \\theta_1 \\right)}}{s} & \\frac{\\partial s}{\\partial \\theta_1} &= - \\frac{r_1 r_2 \\sin{\\left( \\theta_2 - \\theta_1 \\right)}}{s} \\\\\n",
    "\\frac{\\partial s}{\\partial r_2} &= \\frac{r_2 - r_1\\cos{\\left( \\theta_2 - \\theta_1 \\right)}}{s} & \\frac{\\partial s}{\\partial \\theta_2} &= \\frac{r_1 r_2 \\sin{\\left( \\theta_2 - \\theta_1 \\right)}}{s} \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "We can sub each of these into our equations above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "id": "98ff178b-d6bf-4b72-b7e5-0e9ea08f35b8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{r}_1 = - \\frac{G m_{2} \\left(r_{1} - r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)}\\right)}{s^{3}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} + r_{1} \\dot{\\theta}_{1}^{2}$"
      ],
      "text/plain": [
       "              G⋅m₂⋅(r₁ - r₂⋅cos(\\theta₁ - \\theta₂))             2\n",
       "\\ddot{r}₁ = - ───────────────────────────────────── + r₁⋅\\thėta₁ \n",
       "                    3                                            \n",
       "                   s (r₁, r₂, \\theta₁, \\theta₂)                  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{r}_2 = - \\frac{G m_{1} \\left(- r_{1} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}\\right)}{s^{3}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} + r_{2} \\dot{\\theta}_{2}^{2}$"
      ],
      "text/plain": [
       "              G⋅m₁⋅(-r₁⋅cos(\\theta₁ - \\theta₂) + r₂)             2\n",
       "\\ddot{r}₂ = - ────────────────────────────────────── + r₂⋅\\thėta₂ \n",
       "                    3                                             \n",
       "                   s (r₁, r₂, \\theta₁, \\theta₂)                   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{\\theta}_1 = - \\frac{G m_{2} r_{2} \\sin{\\left(\\theta_{1} - \\theta_{2} \\right)}}{r_{1} s^{3}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} - \\frac{2 \\dot{\\theta}_{1} \\dot{r}_{1}}{r_{1}}$"
      ],
      "text/plain": [
       "                    G⋅m₂⋅r₂⋅sin(\\theta₁ - \\theta₂)   2⋅\\thėta₁⋅r₁̇\n",
       "\\ddot{\\theta}₁ = - ─────────────────────────────── - ────────────\n",
       "                       3                                  r₁     \n",
       "                   r₁⋅s (r₁, r₂, \\theta₁, \\theta₂)               "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARMAAAAkCAYAAACqod4ZAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAABJ0AAASdAHeZh94AAANWUlEQVR4nO2defRd0xXHPyFEojElbShKWKU0/H7IKiWGmKqGVgylSMy6qorVGlZEfW1DF2pISkzV0pq6aqaCFk3MgkqIUFPNEgnVRQ0R0j/2ub/f/d3ffe/e9+59v8n7rvXWfe/cc/be97179j1nn32+r9+iRYtoookmGgszWxNA0kvdbUuj0D+t0MyOBpYDJoaio4H3JU1Mq99EOTCzK4Chknaus/3ywHPApvGb1syOwH/DlYFpwFhJ8wobXJttNdtgZtcBj0g6p/EWNhz3hOPq3WlEI5HqTPAffTXgivBZwKu0O5ceBzNbBzge2AYYBszFO9ZVwFWSPu9G8/LiKKBfgfYnAFMSjuR0YF/gIGA+cD1wFnBgAT01oYANpwDTzOwySf9trJVNFEWqM5G0eqKoyA3ecJjZrsCfgZuAccBLwFeALYDxwHXAR11kS39JC+tpW6TDmNkg4BBgl1jZSPz6N5X0SCi7ADixXj112FW3DZKeNrOXgf2AyQ01tMFI6VN9DpVGJr0GZtYC/AU4WdKvE6dnm9klkhaFuqsArwN7Az8BNgFeAPbCnc+5wEhgFrCHpNcydEfyfgwcBmwKHA78oUL9LfAn8gjgc+BfwEGSZoXzVxCb5pjZVGA28H6Q/wXwJ+A4SV8kxO8ILAIejJUdA9wXdeKAecDQatdVMoracCv+/fZqZ1Imemr8ZbHuNqAETAIeS3EkAESOJKAlHH8KGLAxMACfzp2FTxO+i9/ov8ihO5J3PHA28G3glrSKZtY/nHsgtNsYnzZmTb/2BRbijiqKO+yVUm9z4ImY41wCH6XcmKg3EOiSKUNJNkwHvmNmA8u0rZfjHtpjMD0GvXpkYmarAVsC+8fKFgPmAINC0Q2SovOt+E28l6S5of7f8ZHKtyS9G8qmASuF9zcBWwH3SNojYUIr8Amwp6QXYzaktVkGD2rfFnuiPJfjMmdLOim8f97MDsXjQtcm6q0GvJWwbRBwppnFHe0SwJMZtpaFqjaY2arAlcDXcId5qqTrEjLeCvW/jk9fm+ihqLSaMwgfnu6DR5/n4T+6JH3WZdZlY71wnJ4o3zQcH8KnEhFagL9GjiTgG8CNkSOJlT0W3k/Cpy370xkteMDzxUR5pzaS3gvTmLvMLHqyXJ81lQKeSnx+C+98SQzEg84R1gYW0P4dRbiWjlOhatcHgJmdBkzIsHO0pKmJsiwbFgJHS5phZisCT5jZFEn/i9X9OBx73cikUf2op8ZfOjkTM1sJuBv4Jh7QvAXYGQ+iDcFjDT0Fg8OxLeAZYgkvhmXSrwIzY/Vbgd8mZGwAnJQoawEuDfKmmtlWFfS3AhckCyu1kXSgmU0EdgB+AJxuZrtKuquCfIDkTbeI9OnpfGD52OdlgfmJEdPQYPORWbYmMBFfFauGNKdY1QZJbwNvBzvmmNl8YAUg7kxWCMcuXcoGMLMhwBhgJ9whrow7x6eBy4HLU2JXUdse34/MbBj+cLpQ0s+LyuvgTMxsSeA2fMg8WtKDofxU4BngEDOTpDlFFZeEWeE4CkiODkaE40wAM1saWJOOQ/whwKqJslXxH7utLA0xef+sxWBJM4NNZ5rZHfiIoJozyYsngQNin+cDg81ssdgNPx54WNLDtQiWND/IqxW5bTCzjYDFJb2ekDECeDMxmuwq7AlchDu8f+AOcxiwG3AZ8H0z2zMRl+tN/eiH+IMpGdOqC8mRyTHARsAR0RcAIOnDMLc+Cg/0Jee13YKwdHgLcE74Ae/Dn9zr4wHUdyS9EaqvH44zYiJagU/xHzhe9iGdnVMSafIqwsyG40+jW4E3gTWCjIvytM+Bu3AHNSRM2e7Ff98JZnYlsAcwFtisJH15kMsGM1sBX6U6NEXG5pTjbOvB8/gI8vb4CMTMTsCn1rvjjuWGRLve0o/GAO/i/aYw2pxJiJYfi3vhS1PqRjGFFctQXCJ+hK9wHIEPxxficZLbgYtj9VqAFxLz8Q2AWYm8kBZgZqXha4q8D3Pa+RGwFn4DDcXjG1cDZ+ZsXxXBsU7Hg8mTJc0zs3HAOfho4H5gS0kvlKEvp02ZNpjZAOBm4AxJD8Xbm9lS+A3/va6yOQ5J91Yon2NmFwOn48HrNmfSyH5kZqNxB30Ong4xAXdKywMjJD1TpXlS1jLA1sA18YTOIjriI5Mx+GrD7ysEh5YKxwV5De4KSFqAL+uelVHvYjo6FySdjS/pxstOA07LobeTvIz6c/GnWLU6ByQ+b5VVJwEDJpnZxZI+l3Q9nm3abahmg5n1w5fl75V0ZUqVg4FHEzkqPQVRH0kmKDayH20YjiPwuNftwCX4gkGelcE4dgaWxOM5peiIO5OdwnFlMzs5pe624Zic09aE2L6fvJgh6eYiOovAzO7GRyFLm9kb+DJw1ZhDPW3KgKQ7zWwysAq+/SET3WVrwGZ4zsxTIYsZfM/O0+H9Z0DhwGDZCDlD48LHOxOnG9mPoo6+GbBFQSc7Bg90/60sHXFnMioc985oMzuv8Ao4Gg9M5cUf8WFwt0DSttm1ircpC5KSq1VZ9bvT1geokjgpKW2a0BNwBv7knpKyEtfIfhR19KOKOJIwfdwBuEPSJ2Xp6B+EL40PY56RNCJZycwG43O9OZJeCWXj8WH72ngQ8xFgfJQaXgk9dY28ib4BM3uF2h5WV0varwb5RwK/xIf8YxPnGtaPguy1gHdo34AbP19Lf9wO3z7SYYpTVEf0VFg5HN9MUQywPZ6FOCVWthVwIZ4gtjU+d7w7ROabaKK78BIegM/7eitdTGcEGoVJ+KhitKT3ElUa2Y9a8P7aYWWpDjngDmEBHg8pTUc0zVkyHD9NEQDtW8XbNrBJ6hBhN7OxeKr6Zvgaeyp6W8ykid4FSds0Qm64b8/Dc5u2kfROSrVG9qNo+vFomuC8csxscXy/1L3qvEu9kI7ImUTJM52Wq8xsE3xH6h2SkmnrcQzGvdp/qtSBEmMmZtakifsSQlKXUmKY2fF4nGQGsF1I4ktDI/tR1NEfz2l2JTlb4EmZyVWcwjr6g2c4mtmzwEZmtr6kp6BtI921uPc5PEPwJPzLrroSUGbMpCtuKjNbDk+L7h9ekyT9rtF6m+gZMLNf4SRNTwDbp0xt2tDgfrQh7an8eVBJzhicyiJtd3shHf0iDlgz2wdPopqP78NYGk8IWwTslEwoisPMzsWj16MkvZzTkF6BMCwcIOmjEKCaBYxUx42BTfRBmNn+eCDyc+B80mkTXpF0RaxN6f0oJPZ9ADwlaWQOuyvJ6YenDLwmaVSiTWEdbUvDkq4x5584Duf7mI9nwJmkSgElzOy8IHR0X3MkACE7MGJpG4CzznUYEVkF7tWeDGsgL2w9sq1n8r0OD8fF8etJwzRiKx8N6kcj8MDtE1kGZ8gZie9Fm5jStLCOfkXY6c1sEp50NFrSs3UL6uEIU51p+A7QYyVNTpz/Dc6Q1mW8qkVg6ZysD5Zhf72yzWw9/DsenhIY7NMoqx9lyTHnlBkPrCHp32XrqNuZhEzLscCudEzA+bCG/SpdBivAzRqTMQzfYbmb2smVBuFLgbuEJKyG6S9Dnjkn63Q6crIeCZwoKY0npctkm9nj+Lb+Lw1FY1n9KI+cEM/5VFJrI2wtwrQWBZKS9HEGnFxAbmFY7dys/fDNWQfjSUcfAPdL2j1eT9JcM5uJb3yK9pt04l6tVX/Z15OBRvLCNvlea0dZ/ShTjqR1arStJh11O5OuXp6rEXFu1gl4ctL7Veofi3OBHI5TD6yEUxFEo5GPJH1gZsviS2tx2oAO3Kt16s9CKfKsnZN1fOJUYV7YkmRPB040s4GSPs6s3QdQVj/qiv6YpaNXc8BWQSsp3KxVEO1TiDzuq3iqMHhOzKVh9NIPOD+2ES06n8yirFV/FsqS10oOXthulN3ke+3F6KvOpBM3q1UnL74VJ1hqwflGbogSk0KCUWsVXUnu1VT9wYZ6yZtrvZ5KyMULW6edVWXntLfX8r020Tf+6iINrfjKQBwRefG6+B6JiSFvBPnfnq6Nbyc/HHjJ/B8C8yDJvVpJP3iSz7iU8iykyat4PVXQxskavfDpUisdOUfqsTNLdh57u43vtYni6HPOxCpws0p6W9KM8H4O7gRWiJ1/UU6WtBE+nVmffHgSWDdLf9AxFQ/u5ka911MBbZyssbJOnKz12JklO6e93cn32kRB9MVpTiY3q8XIi8O+i7l48G8hTvC8AJiaU1+Se7UmbtgcqOl6MmQ1khc2t+wq9nYn32sTBdHnRiZkcLNaO3nxYaFoAL5K8jj+Pzst+K7QXE/HEIyNuFcz9VeCmR1gZovMbPXEqVqvp6KskIU6Dv9P4tk4r8WWqoEXtqjsNHtDecT32tz31EvR50YmqsLNainkxZJOwTdyFUGce7UmbtgYhuOd8I14Ya3XU01WkFeUF7Zu2VXshZ7N99pEDvTFkUkqLJu8uG5IuhNPtFqlgJgdgZ/lzWrNuJ6aZNWIumTn+P57JN9rE/lRaG9Ob4KZjcL/HyT+d5tx8uKusKGNvBl4jwLkzY28njLtjMns9u+/icbiS+NMmmiiicbi//QTGz2XSuvFAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\ddot{\\theta}_2 = \\frac{G m_{1} r_{1} \\sin{\\left(\\theta_{1} - \\theta_{2} \\right)}}{r_{2} s^{3}{\\left(r_{1},r_{2},\\theta_{1},\\theta_{2} \\right)}} - \\frac{2 \\dot{\\theta}_{2} \\dot{r}_{2}}{r_{2}}$"
      ],
      "text/plain": [
       "                  G⋅m₁⋅r₁⋅sin(\\theta₁ - \\theta₂)   2⋅\\thėta₂⋅r₂̇\n",
       "\\ddot{\\theta}₂ = ─────────────────────────────── - ────────────\n",
       "                     3                                  r₂     \n",
       "                 r₂⋅s (r₁, r₂, \\theta₁, \\theta₂)               "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r1dd  = r1dd.subs(   s.diff(r1),  (r1 - r2 * sp.cos(th2 - th1)) / s )\n",
    "r2dd  = r2dd.subs(   s.diff(r2),  (r2 - r1 * sp.cos(th2 - th1)) / s )\n",
    "th1dd = th1dd.subs( s.diff(th1), - r1 * r2 * sp.sin(th2 - th1)  / s )\n",
    "th2dd = th2dd.subs( s.diff(th2),   r1 * r2 * sp.sin(th2 - th1)  / s )\n",
    "dispForm('\\\\ddot{r}_1', r1dd)\n",
    "dispForm('\\\\ddot{r}_2', r2dd)\n",
    "dispForm('\\\\ddot{\\\\theta}_1', th1dd)\n",
    "dispForm('\\\\ddot{\\\\theta}_2', th2dd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "723f9ea0-6e95-4c23-9bbc-a8bbe756d095",
   "metadata": {},
   "source": [
    "Finally, sub $s$ with our formula for $s$ above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "id": "027d89e0-9235-4f38-9300-5f096defad71",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{r}_1 = - \\frac{G m_{2} \\left(r_{1} - r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)}\\right)}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}}} + r_{1} \\dot{\\theta}_{1}^{2}$"
      ],
      "text/plain": [
       "                   G⋅m₂⋅(r₁ - r₂⋅cos(\\theta₁ - \\theta₂))                  2\n",
       "\\ddot{r}₁ = - ─────────────────────────────────────────────── + r₁⋅\\thėta₁ \n",
       "                                                          3/2              \n",
       "              ⎛  2                                      2⎞                 \n",
       "              ⎝r₁  - 2⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠                 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{r}_2 = - \\frac{G m_{1} \\left(- r_{1} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}\\right)}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}}} + r_{2} \\dot{\\theta}_{2}^{2}$"
      ],
      "text/plain": [
       "                   G⋅m₁⋅(-r₁⋅cos(\\theta₁ - \\theta₂) + r₂)                 2\n",
       "\\ddot{r}₂ = - ─────────────────────────────────────────────── + r₂⋅\\thėta₂ \n",
       "                                                          3/2              \n",
       "              ⎛  2                                      2⎞                 \n",
       "              ⎝r₁  - 2⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠                 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{\\theta}_1 = - \\frac{G m_{2} r_{2} \\sin{\\left(\\theta_{1} - \\theta_{2} \\right)}}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}} r_{1}} - \\frac{2 \\dot{\\theta}_{1} \\dot{r}_{1}}{r_{1}}$"
      ],
      "text/plain": [
       "                             G⋅m₂⋅r₂⋅sin(\\theta₁ - \\theta₂)             2⋅\\the\n",
       "\\ddot{\\theta}₁ = - ────────────────────────────────────────────────── - ──────\n",
       "                                                               3/2           r\n",
       "                   ⎛  2                                      2⎞               \n",
       "                   ⎝r₁  - 2⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠   ⋅r₁         \n",
       "\n",
       "̇ta₁⋅r₁̇\n",
       "──────\n",
       "₁     \n",
       "      \n",
       "      "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\ddot{\\theta}_2 = \\frac{G m_{1} r_{1} \\sin{\\left(\\theta_{1} - \\theta_{2} \\right)}}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}} r_{2}} - \\frac{2 \\dot{\\theta}_{2} \\dot{r}_{2}}{r_{2}}$"
      ],
      "text/plain": [
       "                           G⋅m₁⋅r₁⋅sin(\\theta₁ - \\theta₂)             2⋅\\thėt\n",
       "\\ddot{\\theta}₂ = ────────────────────────────────────────────────── - ────────\n",
       "                                                             3/2           r₂ \n",
       "                 ⎛  2                                      2⎞                 \n",
       "                 ⎝r₁  - 2⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠   ⋅r₂           \n",
       "\n",
       "a₂⋅r₂̇\n",
       "────\n",
       "    \n",
       "    \n",
       "    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sForm = sp.sqrt(r1**2 + r2**2 - 2*r1*r2*sp.cos(th2 - th1))\n",
    "\n",
    "r1dd  = r1dd.subs(  s, sForm )\n",
    "r2dd  = r2dd.subs(  s, sForm )\n",
    "th1dd = th1dd.subs( s, sForm )\n",
    "th2dd = th2dd.subs( s, sForm )\n",
    "\n",
    "dispForm('\\\\ddot{r}_1', r1dd)\n",
    "dispForm('\\\\ddot{r}_2', r2dd)\n",
    "dispForm('\\\\ddot{\\\\theta}_1', th1dd)\n",
    "dispForm('\\\\ddot{\\\\theta}_2', th2dd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97f3552c-f58a-438c-894a-3361a84e3d4e",
   "metadata": {},
   "source": [
    "Create a vector which returns input state and updated state"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "id": "93f6e5b9-b5bf-436f-a7d7-8e9a55e7bb39",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Input State:\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{1} & r_{2} & \\theta_{1} & \\theta_{2} & \\dot{r}_{1} & \\dot{r}_{2} & \\dot{\\theta}_{1} & \\dot{\\theta}_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[r₁  r₂  \\theta₁  \\theta₂  r₁̇  r₂̇  \\thėta₁  \\thėta₂]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "State Update:\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\dot{r}_{1} & \\dot{r}_{2} & \\dot{\\theta}_{1} & \\dot{\\theta}_{2} & - \\frac{G m_{2} \\left(r_{1} - r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)}\\right)}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}}} + r_{1} \\dot{\\theta}_{1}^{2} & - \\frac{G m_{1} \\left(- r_{1} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}\\right)}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}}} + r_{2} \\dot{\\theta}_{2}^{2} & - \\frac{G m_{2} r_{2} \\sin{\\left(\\theta_{1} - \\theta_{2} \\right)}}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}} r_{1}} - \\frac{2 \\dot{\\theta}_{1} \\dot{r}_{1}}{r_{1}} & \\frac{G m_{1} r_{1} \\sin{\\left(\\theta_{1} - \\theta_{2} \\right)}}{\\left(r_{1}^{2} - 2 r_{1} r_{2} \\cos{\\left(\\theta_{1} - \\theta_{2} \\right)} + r_{2}^{2}\\right)^{\\frac{3}{2}} r_{2}} - \\frac{2 \\dot{\\theta}_{2} \\dot{r}_{2}}{r_{2}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                 G⋅m₂⋅(r₁ - r₂⋅cos(\\theta₁ - \\theta₂))       \n",
       "⎢r₁̇  r₂̇  \\thėta₁  \\thėta₂  - ─────────────────────────────────────────────\n",
       "⎢                                                                        3/2  \n",
       "⎢                            ⎛  2                                      2⎞     \n",
       "⎣                            ⎝r₁  - 2⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠     \n",
       "\n",
       "           2         G⋅m₁⋅(-r₁⋅cos(\\theta₁ - \\theta₂) + r₂)                 2 \n",
       "── + r₁⋅\\thėta₁   - ─────────────────────────────────────────────── + r₂⋅\\the\n",
       "                                                            3/2               \n",
       "                ⎛  2                                      2⎞                  \n",
       "                ⎝r₁  - 2⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠                  \n",
       "\n",
       "             G⋅m₂⋅r₂⋅sin(\\theta₁ - \\theta₂)             2⋅\\thėta₁⋅r₁̇        \n",
       "̇ta₂   - ────────────────────────────────────────────────── - ────────────  ──\n",
       "                                               3/2           r₁               \n",
       "   ⎛  2                                      2⎞                       ⎛  2    \n",
       "   ⎝r₁  - 2⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠   ⋅r₁                 ⎝r₁  - 2\n",
       "\n",
       "    G⋅m₁⋅r₁⋅sin(\\theta₁ - \\theta₂)             2⋅\\thėta₂⋅r₂̇⎤\n",
       "──────────────────────────────────────────────── - ────────────⎥\n",
       "                                    3/2           r₂     ⎥\n",
       "                                  2⎞                     ⎥\n",
       "⋅r₁⋅r₂⋅cos(\\theta₁ - \\theta₂) + r₂ ⎠   ⋅r₂               ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "S = sp.Array([ r1, r2, th1, th2, r1d, r2d, th1d, th2d ])\n",
    "Sd = sp.Array([ r1d, r2d, th1d, th2d, r1dd, r2dd, th1dd, th2dd ])\n",
    "print('Input State:')\n",
    "display(S)\n",
    "print('State Update:')\n",
    "display(Sd)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acf83d9c-d42e-40ce-a6e8-820feffb929f",
   "metadata": {},
   "source": [
    "We can convert these equations into a function which handles states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "id": "c8e494f4-8503-4015-ba42-86ea2df084d0",
   "metadata": {},
   "outputs": [],
   "source": [
    "def generate_system(m1v, m2v):\n",
    "    \"\"\"\n",
    "    Generate system state update function \n",
    "    based on masses\n",
    "    \n",
    "    :param m1v: value of m1\n",
    "    :param m2v: value of m2\n",
    "    \n",
    "    :return: lambda function\n",
    "    \"\"\"\n",
    "    # G constant\n",
    "    Gconst = 0.1\n",
    "    \n",
    "    # Return function\n",
    "    Sm = Sd.subs(G, Gconst).subs(m1, m1v).subs(m2, m2v)\n",
    "    Smf = sp.lambdify([S], Sm, 'numpy')\n",
    "    return lambda t, S: Smf(S)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1fff0ac0-f3b0-4979-b276-4e0c6357062e",
   "metadata": {},
   "source": [
    "Plot output of system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "id": "d785cdca-ef29-40e8-aa57-ac6fcfe25883",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "deg2rad = np.pi/180\n",
    "\n",
    "orbit = generate_system(3, 4)\n",
    "solution = spi.solve_ivp(orbit, (0, 150), np.array([\n",
    "    3, 5, 5*deg2rad, np.pi - 2*deg2rad, 0, 0, 2*deg2rad, 2*deg2rad\n",
    "]), max_step=0.1)\n",
    "\n",
    "r = solution.y[[0,1],:]\n",
    "th = solution.y[[2,3],:]\n",
    "\n",
    "fig, ax = plt.subplots(subplot_kw={'projection': 'polar'})\n",
    "ax.plot(th[0], r[0])\n",
    "ax.plot(th[1], r[1])\n",
    "ax.grid(False)\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.16"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}