{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sm.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Define variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "m, M, Ic, k, b, l, g = sm.symbols('m, M, I_c, k, b, l, g', real=True, positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = sm.symbols('t', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sm.Function('x')(t)\n",
    "v = sm.Function('v')(t)\n",
    "theta = sm.Function('theta')(t)\n",
    "omega = sm.Function('omega')(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Kinetic energy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle l^{2} \\omega^{2}{\\left(t \\right)} \\cos^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(- l \\omega{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + v{\\left(t \\right)}\\right)^{2}$"
      ],
      "text/plain": [
       " 2  2       2                                   2\n",
       "l ⋅ω (t)⋅cos (θ(t)) + (-l⋅ω(t)⋅sin(θ(t)) + v(t)) "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vc_squared = (v - omega*l*sm.sin(theta))**2 + (omega*l*sm.cos(theta))**2\n",
    "vc_squared"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\frac{d}{d t} \\theta{\\left(t \\right)} : \\omega{\\left(t \\right)}, \\  \\frac{d}{d t} x{\\left(t \\right)} : v{\\left(t \\right)}\\right\\}$"
      ],
      "text/plain": [
       "⎧d               d             ⎫\n",
       "⎨──(θ(t)): ω(t), ──(x(t)): v(t)⎬\n",
       "⎩dt              dt            ⎭"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq_repl = {theta.diff(t): omega, x.diff(t): v}\n",
    "eq_repl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\frac{I_{c} \\omega^{2}{\\left(t \\right)}}{2} + \\frac{M \\left(l^{2} \\omega^{2}{\\left(t \\right)} \\cos^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(- l \\omega{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + v{\\left(t \\right)}\\right)^{2}\\right)}{2} + \\frac{m v^{2}{\\left(t \\right)}}{2}$"
      ],
      "text/plain": [
       "     2        ⎛ 2  2       2                                   2⎞      2   \n",
       "I_c⋅ω (t)   M⋅⎝l ⋅ω (t)⋅cos (θ(t)) + (-l⋅ω(t)⋅sin(θ(t)) + v(t)) ⎠   m⋅v (t)\n",
       "───────── + ───────────────────────────────────────────────────── + ───────\n",
       "    2                                 2                                2   "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = m*v**2/2 + M*vc_squared/2 + Ic*omega**2/2\n",
    "T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Potential energy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYsAAAAvCAYAAAASGubHAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAQkElEQVR4Ae2d7ZEVtxKGD1sbAIYIrskAQwTGGWAcATgDu/jHP8rOAJOB7QxsIgCTAb4RmLsZcN9HRz1oNJoZnfk6s+vuqll9tVrSK6lb0mjO3vr06dPByRFwBBwBR2DfCLx48eIn1fC2ni/1/K3nR8Vdyd2ELjcpxQtxBBwBR8ARmIxANBSv5GIkDnJ/k/NGz1eEt6CLLQrxMhwBR8ARcARmIfA4y/1S4fsyGuwyNiE3FpvA7IU4Ao6AIzAbgdQw2PFTGje7gCEBfgw1hI6nOQKOgCOwAwS0g7iXVeN+DL/L4ptg3HU8kvtLE9njEQ87l/dywzFXie2Wv+AuweJxjoAj4AicjoCULS+geZ/wQM9HhXMlf7rQQg7J/aBo3mH8XEg+KD7UQ+43pfRSnHipNy/NiwbDj6FKqHmcI+AIOAITEJCivYoKGoX75wQRo1kkn1tRf8otGooogJff3/cJU94PevIjrKfix2AUyY1FERaPdAQcAUdgFgIcE/0xS0IhsxT8M0XfljtkCOB5J57iDkHxHDl9macrzHsQjNAPcjvk7yw6kHiEI+AIOALTEZCyfRRzL7qziHLvyQ2GIoY56nqf1fZHhYeOn0jL85gIbln9V09n1+I7C4PIXUfAEXAElkEAZfy3lLjdWJotVbLYqXD89Fb+xzzyYxRauwfFh6Mlua148aWEMSsaMuWjztTdDF6Tz3cWDRTucQQcAUdgEQRayliKF8V+Rw8f0P1kilwuL6E5MoIe6kH5o+wxDHf1/CMeW+HzDgL+1jsFpecGibI6hkB81OlbPcgPZSjulfy8u7AyFAxEfnhbctxYHMHxv46AI+AIzEZAiheFjrLnOOegMOf/v+tBQaOc/9LzS+R7LhcDAR9Gg3cc4YaTwtx2whAERa7wF/LXELualkEhk/Kj+HkfgTHhOu3QMRVld96JXCjSyRFwBBwBR2AZBFjBQ/aiGMPAkRDfQ3wvv33z8FzhYFDkQlyxxaBYOsqa1f2phIyPA5mG3ldYNvJj9FrkO4sWHB5wBBwBR2AWAihjdgQcNzWrc/mJM0NAAS9jHH6I3QgGBr6D3NYREHGVxHFXkNHDjzFjpzNEGDeMTot8Z9GCwwOOgCPgCMxCAGWMouddwB96itdQFZ8rdPItftU2bYnKvK0wRmBSOW4sUjTd7wg4Ao7ARAQSZWxfVrOzYIeBIeglpbOrgKbuJo65j3+LR0iR4QGuymvKkR8DkhO7k85tKjcWOUwedgQcAUdgGgJPyJYoY3t3EBSy4rn6Gkh+rr+aog7GROH3SfoPSbpF17goeZR9iVrvK6iDmNhp5ETcVR7pxiJHxMOOgCPgCExDAGWcrtpN4bZW6VLSGAduLNmOg3wNKT0oa7mWv0mr8GBw+v7HBcYp1EWy8fMVd2OgEtnkb9ph8ZfmucluBJ/rYukLpmvV5LE2KJ3OZ2XDF529L7DG5KSgiJeVx+AvUdbwpDL36HdM9tErS/fDGVrFHOR6bEpcjeUWFNdR7fYTN6OYoyhr3mlw6+mZ/Ha19iD/VF0Vrt9KXonY2XBEFr7tkJt/X2F5OK56agFzF/3V2VgJrCRKBvpWcb2KCwalAxaNuNIDiDRmMI94qkmy6MDf5ObWm3I5U8SKf6X0koVV0vmprw1WM6WzQmFQMujwf6O45iaGwoHG5Bhf6ioPK6DeX6KEt4Ynlbknv2Oyj95Yqx/20bptayEsMUzogNaOpqYWsR/+ktv5tdxFj6FUAHeKUVhmFVHEvSReXuw8jAwYFhq4mKGIct/ILSlOrGo4Q1SZuzUUQ20gTXUHQxQ6+PGLl+DHKgUjmVMRC2NSnpN/iTLmZRXS+RDI5O7cdUz20UFr9cM+WrdtLdBr4WO/CcXy/Ue+OwpiFjUWSJTCwUDwlSJWzQyBvEViuxNI+TpnZJY21ZXMwV9flNzi2dzU8tbIV9EGW/mzM0vJzkND3JgcpbMbPPmXKBGuvJRtHyERNYmogx52fJuQyhocH9RHFTkrJpsAceZC1uyHmqbRz3o2G3c1dZrDo7awWGfcDi7W8zLEzwLzkdzi8dTixoLC9KD4MRa9lVWFmKi/6jF+eRcnrGvYPfRIfqL4SXeOe+StEd3bhojhHbm2k0vLz29E9MqJmTim69thcdbKimOIaniG8p8jzTE5B+rdMtfuh26JNz+GE57iDmGg6eGEoi99DWPB+T+GotdYKB0jQjoKDWu2uMKOZRxiXVREm2I6ZS++o2mXND001gZJZpJhcBtSHo6loI9Hp9ntHZQG5n3Ua7SVj51D8ZcoTVgNj/HuwVV9w0LGMTlvb2zRD+dt4XlKj/ORF+vhZfZYLSIf/L064nJMyIz0D8qLMi4R276fk4aMKmzxciTA6he5EI2ica0X1yHl+Af+Ibkox4Py962mgxSl0wZ7AU8cYb7MbN6tRB5W3v/AIOIXI+FpypcfJU6ZqZH8TvF919zEGi4KNDKIMFI+ZKHw8ncFoV2KR8EbFbGIMliBIIfnvuJYjVT/EqV4U6KuyCvWOWVc2q960y82MTj+xJCGNsmlP9Jf8FSwjO1NwoRGQmrT4NyJbWYegRe4QeSBwJKfxWa+Moa/08M4J57LKK2+Vniv/aDq/rtIfYGuKZ06dIAQ7yjfZSfXjAgVyGDjfQUULJTiUECNQpafQWgVC4o+TQ85sz9KR1kjx/gpB6NhcrIcIQhvrkhTPtJbAz1NxK/yrD1P5Q/GQS4T5o3ccKU0himndftA8RgL6swkYwLxJWeov/yBFO68eLe06A61wfJiMM1PtmAsFJe2rSgn8sz6JcqsvvRJWpcseZ2g2gG+z+UGRScXoxGuEMoP/tTrSk96FnujMVFbA6ntNXOHscuNN7t2ydgNWMkF2//JRR5j3jBmHjPum19Djbx77Qfq7zQDgUWNheqBojIlZQYChRv8cTAd5DJxoZT/GJP9FS+DkpdP6aDkSATOoeMryv0IUw9R9sueNItmMlBWs4tQGLkQkwiC53fxBOMYYo5/mFRcQQOPO3oeyM+/Q7S2wzV2pjjUBowWk5eVfEMKf1LAsLf4ITnwoDjzPJbXXLC0Nltc7tbw5HmWCLOrS/uy9Aueef/cdEwOGgujc0c8jKO3sRPAhBt1qVFl7EJ3FW9zmzDx+XjYcz9QZ6cZCFxqANDhb06UwUq7pFx4XxFW+3JNoTMAjZ4p3lYsxFP2kMIn32s9za8xEiEZDHAoHbzHmM9/GcxXn4OffTE/ZadG4DODfOIhnXJauxfFkyfdZdAOm2zyHkl8KHICHDWxakORskKjzrQZA5NOSkV1qLcN4qTcVn7JM1xSxYnQITmkYzh7sYBBhLJN+zJEZn9qeA6qJ6tdyswJzO8onaOOnBhPLcOYMLxUWtrX4NCMGaWVxsneMKHtS81Dg6Zm7oArY5Xy6V8WDin1jSn4clz33A8HtXHpcdfgJNks0m40YSyYZEPn5nMAQHb4uEPlsMpJFZIpi3zANeUpDwOVQczqPSXyMsiRP4XIzwoqX22mskwxcoTRR8YzVA+bbGDMyouyGbQcS/FdysnHNsoDJlBupJBl31oEhrE/URbtGDPaY6Kq01VmOMrIMyieMcKVv5YRzPnysPhz/MG4WEaetxSWPPDdGhPasNg8VBuq5k6CHZgdFM7nI0ahNNfgby1KElmIgnbVD6pfcUwoftK4Ozbx+FcybqXhm+i/WKpRAovJZe8rTGxYaSqNyceKMVXODEIUW2mHYvkfRE9pAOdxlsfcoSORZlUU62Z5Utfq2vmSMWEyHtrXR0w0sDnIZYfB7ouBhWJ/Jn+YpKQXaKgNsKNgUuKsvjWBY+KQnICx6tHgKX+pPXcky9qblpn6a3hS/sX9qrsZ56Y9PYXcdExOnTvMifcFrBif6SLvIIyJY4zYDrszXnbYD4WmedQpCCxmLFQoAyifoO8Ux6BFKbaOcyI/6UPEhGZw5kqKssZWwuRBeZUozc9qv0MqE0XM5LFJ1+JROje6SIePidYipZnSYleEsbCbOoFP6eDBZAuGJER2/xTboLyUydOQ4nivg2EqrcqLcmLmlpJQ/seKL9WJuFaZMX/q1PCk/Iv4qbOe21EYfXtQuFF88pd+wfNGYyIITp07fHPUmr8RU/o0n2scB3J0ZQshw3zP/aAqO81B4GJO5iwvq2YmYEoc4TCJ80GIIiU+H4SKalHIJ7kM2EDy25FUS6alJy7KYmhbz0CnDnbdNcnaeJkUHIvkip5jJFNGX8v/RDxmHCzza3m4iWP15JaIKTTj6WBjCdEdasOv4gllSi74sFPpGK0KOdQh9FusH+21tsXswQFLa0san/preFL+2X7VFUXFmAgKS24Lg4gNO9irrLAhbK81JrGd1XMnYkSb8/loRiDvdxZQFsdtPN6/7b0fsu734KkIXJ6aIefXILGJelt+jAN3r211y4TEiATlI5dVK0rNVusMtIcK88I8n8wH4vQw+Tnffyv3rh4UG4q+w6/4lBj4r9KIxE8dMAThymAS3/KqDMr5jyJfy0UR0j7qQBtNwbLCIo12Wp2YeIRtQikYXpRjVEiDj+Mt2pUbWEU3NNQGzl/Jz44CWRxvWfmNgOgZkoPhoz3BIMq1vstl0GdP88gsXMOTZZkdZHcadmgRC/qVnSx9H45F5c93tRR6kzE5qM2nzB0WG4z1dLyCEfEl7DjqZO4wZmyO7b0fVFWnOQgs+quzcypSm1cDlEnOwEbhD5J4UO4M6iGFPCjj3IlLtWGOHOXFwBV/idLwqeEx3j5XMlhMnPyCu0/eWPx1wGSsDaekq73Vc+cUuXN51+6HsfptPe7G6tOXrnoyD1ncQXba0izGj9Hr/b1YT/R8yQKndbQTwbLtbk0BAFu8AVGTeSc8S7Vhjhze69gKsg+WGp6+vBbPyra0krX0pd3rgMmkNi8wdyaVOzHT2v0wVq2tx91YfTrpUfdxwsFpDA8nLpzYsIgLx4WdTAtH7NZYCAAGEEBgTY048uK6ab5dtvSWKz4UDytVs8Kt9OsQWKoNU+UoH/g/ktt3PHWo4anBWnJK7xZqsk7iUXmTxkdNe2t4JlW6IpPKnj13KopZjGXNfqippMrfdNzV1KnAQ5+2joFVbxbCV3rQi6vT7HcWK9aQLTNK3s75OZfnbJ3z6VOIM2zAxBJfV1qqDVPkgB35hqiGZyj/OdNuIiZLzZ0t+2WtftiyDWuWxW01dhDowZRYOHMLjUXxqsft1+6dRYpSrR8gxcvqeMsjjtrqVfEt1YZT5IiXF5h8Cd07CGt4qhp4RibH5IzgJ0Uv3Q+J6GvvFTZc1uBY/gv52U0Ekp+FGu/6uOASLhIdU5b/+68wFsvD5hIdAUfAETg/AjIQXOJhMdwyImvU7GINoS7TEXAEHAFHYF0EZCg4lsJQ8D1Xs9tYq1Q3Fmsh63IdAUfAEVgXAW4o8kHkJjc+/Rhq3c506Y6AI+AILI6ADES4yi539HuzpQr3ncVSSLocR8ARcAQ2QEAGgl9tOGxpKCjPjQUoODkCjoAjcA0QkIHg5tO91FDIv8m3ZG4srsEA8So6Ao6AIyCjwNXZh6mhiKhgQFYnf2exOsRegCPgCDgC8xBg9yAJfGyZfpTMryvc0XNf6fnHeopelvb8BfeyLXVpjoAj4AhcXwT4+A6DEd5XZM3o/Wg245sV/D9sPDjOWw74agAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle - M g \\left(l \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + x{\\left(t \\right)}\\right) - g m x{\\left(t \\right)} + \\frac{k x^{2}{\\left(t \\right)}}{2}$"
      ],
      "text/plain": [
       "                                          2   \n",
       "                                       k⋅x (t)\n",
       "-M⋅g⋅(l⋅cos(θ(t)) + x(t)) - g⋅m⋅x(t) + ───────\n",
       "                                          2   "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U = k*x**2/2 -m*g*x - M*g*(x + l*sm.cos(theta))\n",
    "U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Damping"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{b v^{2}{\\left(t \\right)}}{2}$"
      ],
      "text/plain": [
       "   2   \n",
       "b⋅v (t)\n",
       "───────\n",
       "   2   "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = b*v**2/2\n",
    "R"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lagrange's Equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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p4trQbPXewG7L/3P060MrwkU1p3ufjJDSaVcF8pF0NMLsq9qUhvwYHF82jUg2zDKVezVp4INyKQ0dDrQ0AA/LUwyqnau+5MeQHtoDPkTaGAEjcBUIMGNdKgx0pk/1dP77oqP/ITyEfTmnGaVlNQPi2ixWyQSBk3x+LyNG/Ax03HZEfUjP/l76MPpaWUm4RbCmzgx2Swx59G1HIc/Etyk/glxtOw/pO9hWCkK5OevRUejkp/8PRegvclNfLtnIB/lXCucmJ7ar0udzE8sPeu7lDkNZx8oQtEP2EB7/VsI0RipvxhvGqgPlp2E+xGdLXKdg3xRpc2uojltkVvvfkOea/8RQmbdsl6nvKHluaYbq5/e3izR9G/1JMvpP0xbPZdf6uCFct3xvtvw/x7v1tqniptaTPm4CMEBdMrAyoMWA0JcFFUoCv+zo9DmgRYWZcWqNynLfeh4cdNgdGoJFBx8GGJSNF7L7OnbIESjo/G2MgBG4MgT036d/oA9juyT/dQTSmDFmIEDgxr6TTX9BX/ApfhmuSnwumwOgZd+TCHp+oKVPGzIxOcLFENW96wOJXyoO4ZUZcMrOth4ELPh8owfDREypPD3ETPulv+/ru5OgL/5pJUB24FlyBscvy8Dcr7SU+08KeyWbAZM2+lAP7RQKCwetiaOtoh2eNv4YM+S9o76fi4Y46Nj6stWtg0N4pK22ypf2IE/aI8opb8cM8dkMV+U4in2nVNt6huq4Nqe+/83a/8RQmbdsl953VO8M/2n+v0nmkp++i/8PfUkZTn0JBw8MSgoCL8p5zQzVz+9vhpgwpJ9hYoZ2pz+6IywjyZ1DuG753mz5f37WVKCvj8rrN9v96N07Lms6NA2gdJJ0kGnQPaSqh4ieASFe9jqRQkUD6K9l1zS/oXQ0NA1fTadwBkOuUmMgqhrFfaYIBpzanstqGgcaASNgBPoQUF/CAMMZjN5+py/tnsJVfvrX8ha8SUVUWgR6Vhyupl9dg0cO2ho+U3CdQpOXp3QrPWMiq019SmKZ5MCvtIvfnQNmJwpYU2al3f37vqZ+eROs4TMFpyk0eXlKt9Kvfn9LnkP+veNRll3ljUn+D+TeXKF4XGaY+UNLnqtIkK62alAegiQrOp5Z/Ekkg4Y1BAZlyGeuSFMaZr/GZhHLNPYbASNgBPoQoC+7hj6FSZ6+Gbm+ukc4M6lM5lyTWYNHjsMaPlNwnUKTl6d0M2auWdWC35o6luU5lX9Nmddifoo6rqlfXr41fKbgNIUmL0/p3uL9LXkO+feOR1l2JnjYjjskO5dpJvuHlAmW3sYE8lpGnFPopJMfXm8qxBy0TkvalbhFQeLHTAEDOrOENkbACBiBUyHwWhmxrYo+6GKNyo9AyQz1LMWoqTfbSxfPbO8RtKV4lHVZymcKrlNoyvKUfvFYLWgsrWNZllP6l5Z5C8xPUc+l9SvLtpTPFJym0JTlKf3isfr9LXkO+feOR6XsUybZK8mmBVWVCYGUViXEYolAXlvehl+Hl/KYNVAV1Xkrf9+AzR6zO/FvFRq5a7SsbmyqyJCvjREwAjeLQPQ5HIy+dMMe7O9nVoIV6b692zNZ7Y58CR61SizhMwXXKTS18hwjbEkdj1GOOTyXlHlPmI/VdUn9ajyX8JmC0xSaWnnOHXYReDQyMJP6HTl8S/CqyoQyiIPLMTge5KnCcd7g7/FkBGuvastY9TpRAlAGaoayt1unVL60j65CiDJzXwl3kBEwAkZgNgLqa+hP6DOj/5zNYy8JmrpwED0dth4rV0MH/VVO0MzFow+vuXym4DqFpq88xwifW8djlGEuz7ll3hvmY/WdW78+fnP5TMFpCk1fec4dfkF4xAJBr0y/FsvqAWwBxFkGlAI+AHRgmngGDm5OQtvharYPIJTNPjIGIJbKWRFglo6bCdA8CWPAJZ4D1Iv2ZzZ5sJ3g4JC3wphN4xDk3/SQ/1eyD5bdGzqWxZbuDRZrGyNgBIzAewTUn9C3MdGS+sP3MXYZASNgBIyAETg9Ao28O3gp0dpStSsTygzhnGuxOCidtiA1fpSD1igMYZ1r/ZKGI5tVgL+0BA9f/+R8BDdBseWI26C4dQmhHcWCwXaL6wtD0xK7jqG81IV8uC7wQJFoqCnbvxr3TVrCBoURnC7WjNVB8eldkM0K1aAZ45Unhh/0eVjuHovPaffsNiazW+ffSsE7N/q+zebsBEbACBgBI2AE5iOA7M21wkcz1ZWJodw0SP6heO5fr17LOpR2yzjlv/gKOqVlxeJk1xcqPwR2tj6EgMGqySB+ikcZQzFiJQfljOt2B9OIZrIRLzBAeexsyWjyZcUHQXn2tcCTC7ABYV8dgrXiUThRYtnXiJvrLg9WsxR+N8YLmtIoDattrLBVt3aMxZf89uY3JstaRLjx3/1C9kVfEbus9k5lBIyAETACe0GgGY/4HslRV8vblYkZFUcIbc8kzEi3NSmC9tItSmuvIJtVFzUiX2dFoI1tXb0z2jAWLVvHnuOWQfFACN5MkXhge/ez7APBWvmwkgO2d3LvoZ0pSp+p1gFilR0M06Euue/1gB9b3nh/a6aXF8RK95uest1i+16NH2Fj8X3p9hJuTBa0hN4T/kOds2ML2DiJETACRsAIGIG1CCDvLpWVJ+e9RJlAwOwIVRo82fp0UqM8EcwZsDtlGSuE6BEmX8ju2/40yELp2N7CzOMsozSUky1kzGKHotDHgy1YySjd5gdmxJOVkjeyqzPqimNGdfN8U4U2+plQh1g1uC+yPNgeN8ZL8awo8a518JIf3pwbqr4PY/FFuXq94rPonetlOCFCeQ6+I5RJbE6CyTnqPwGiMRI671djRI43AkbACBgBI3AMBDR2IptwLCEmshdnMzYOP17AmRl2Zsu5yYmZXgp7dK2np5yUZa4ik2are/gdMxghFgEdgbRXAQJTxbPvOujl3NzQXmn1oYfz5wo/2hViPXnODe6tQ4Mhh/Brf6DaLWC9vJpCsRWsb5WGfYho/n1mLL4v3bnDjcmKFmjevbeyq4rmCtZOagSMgBEwAkZgEAGNPciZyCbIyUc3s5UJFRAth5ucvtPD9h3scvb36AUngyZfyoIAPmoaOug7M8yjCbch4PwB+fYqE4qn8YlH4GUFZXOBvsnjrimLsuiaJp68d7syMVYHlR1BGIWsNUrDtifM2wfr4XcCLwh7FTul593nP3Gw4kHCsXho9mZU5qTsyh76n9wUJgvbiE6c/ibevYVsnMwIGAEjYASMwDQENOaE/Mj55qFxfBrDCVRPJtDsmqQBqjYDfVBu0U6iO0i4bcBvYkdD1wzbWVDOQjkaFehFy3YTZs7hi+HFQYAhrGagH+KbhGKl75uJTzwVTx3igDhh6eVVeHu2o6FBM/4dApkP9fyk8DZ/uRG0yDNXosYOr/bWQfzghTDMClRuUr0UcJ8Hyl3l1fBBGIQXD192ZxWMsxPlFjnqA21bL7lzMxaf027uVnlpm3innsuNspXqJJs2+b2o09Vjojof3QhTzurwP+Sd59zTSTr1o1fMGRgBI2AEjMCeEUD+4YryVh47dmGfHDsD80+z0whunJfAJIFCjYxw2grsciPAhbKTFIE8PqUsfhSPMA+foCcflIrgU6RIXmhLQTunI75PKE50yi/qg9abXlbZKAV8b+RXPczU4yefjhClcAQryozSlBQS2an8ibl+5D84GB5xjT1Uh0iLQhVukiVlQmFl3aq8GjrOQ9AuL2R3ytiUIywwz/OK8LDH4oNuc1vlBmNuckCBuJONUsGKF7eD0QaUDQUrV5CuGhPV9WRG+PJfAE/awcYIGAEjYASMwNEQ0HgTclUp6xwtTxhbmTgqvC1zBNlo2FAgEMiTu2n8O9kIdZic/iGk+BUtQi77sdvrvhSG4ALl0PYo8n0LUY8h75c9cRGMkkBeudYLX0wITdD8SJlS6PsfhFqu5QWPZ3o+lZt7+aPuULICMGSG6oASg0LDSkFr5H8nT2DfhssxxAs6BMFaOuLCgGfUO8Jyeyw+p93azcpQ3p4fy0+dQ+FECSrb6NoxUZVPZ/TulfieLnPnZASMgBEwAjeDQCNLhbx5sno/OVlOF5aRGoRZfwTr0iA0PlP8F2WE/AjYHSG2oeG8RBLeZIfAj8AW5iuFp5lh2YSTx5BCQLpXepg5b4VwuRGkMUMvEgJ8myZRNz9NevLOlYSc5E40xJNPCKMpXuGkyVcpqMfrFJn9iA5BnxC2MvGNBgTtP2RTZuqMApLPkivowPTWQZTk20kvfoFLLlQH0yFe0PAO9OLRMEFYzNuzCW6tsfhEqHJu+c5F5i/FN29vsGjfG8XV3pWzYHKk+gcOto2AETACRsAIGIEBBJaOw60yIQbM3N60EQaPAgC507aQ8IetcFYEuBKzI7BG/EQb4Y4Z4ruGXy6shgJTE/ISe6VBIESoZ/Y/N6RFWcmFxzx+zE169nkPzaSG0Mz2mD4TNEPloA4YrqFl9py8EabZ58fBfmbMZxmlARNMqcTAK741kQim/DT8qMuYYjeF3SiN8tv8nRPPsg3AuZrPaAFFcExMtqq/+Nx8XzalLU1jBIyAETACRkBj5mrZ90nAmDOLMNvrERCuCKNxXiIYpplqxSH8ssqRC+9sq0HwHdpa82nDqFQ4SFuGNaStNbTlpk1P2fSUgihMoqxJGWq5dh1BE8J9N/bBh9IDNlyf2Qq3crOnn/38fJ27ry5DdYB7WW54tnlAkJkhXgnnvBxy13BhJj/qnLFunWPxLeExHSp7KHB9uEb2F42J6tl2jFEh20bACBgBI2AEjMBxEHh8HLbmmiHATHApvL1RGIIq25s624UUBj3xQwZh705pSwGWtGOz6KRBuK2ZPD2rBQdGeSKo/6onFJoOjeK5kYp46FBOOkZxIdCyqoIyEbcMJTrFgwcrNcT1mWodlJY8O4qEwjhXguLSt5JU5dVkTPmpSzLi8ZkctXIR1sn3IUX7OxbfEm7toMx6njZ8ad87+fM68b2YiI/srxqTqKRtI2AEjIARMAJGYD0Cj9ezMIcRBDgXgHCWG7YIIcB1lAzRIWgTPqYQpHSibwVbuWPLU4eneJUGQZKtRX0GwZsyxHWuNTrOhbDVq1QE2KYUgupf5P5cNKE8BJ9XcnCLUJSTm4ZKYfYAm0jc2EN1+LdoUp7iCz5scTpQaho+WEO8KEdqu6aM1Dnql7EY/WI4eEd983RHdausKA+8F0mJkN3BocGHVbD7oiBXi0lRT3uNgBEwAkbACBiBlQg8WZneyXsQkIAWQhzbYlAe0lWcDTnCGkpGEkxlM+ON0Buz/VxryvcAuHq1FPTuCNODYMj5gtey+VYAQu+U8xIoKn23JVEGFAWuc+3bFnSnOPL5k+heyUZQpn6UgTqG8M1Ba+KoZ9QB4Rx/LlizEoHSQRx0bJ+iXqUCpuDWDNWBcpOeFQl4cfg98m8ZZI4hXihH1CkpTbL7Vjdoty8znqVzLL6k38rPClda5WnwoG1ZDaP909Y7ucuVMfK+Zkyon40RMAJGwAgYASOwEQKP3r3zWcU5WEr4QvBfewB7TpaTaFUuBECEfBSCQSMahP/O9x8GE+wwcss6rOGltChBXHVbPUMyFj8FWvE4+TunPBe/I2N1HosvMTlH/csy2G8EjIARMAJG4FYRGBuHz7oy0QgVzP5iYstOO2P/ELy733xW/SyFE2589K3dbtPgyFaWzjaWgcKBOTP4o4rHAI9zR21ZhzW8OFvSt9IDRmPxU3A8xzu3J0zOUf8p7WIaI2AEBhC40DF+oEaOMgI3i8DgOHy2lYmmk2GbDFsvkpEbAYbtKcyaDxb8IcXt/WYYfSB32r4je/KqRCDWpGE71dB2oiDfpb1lHZbwUhpWJfjqN9u5DsxY/EGCnQUYk501iItjBC4Igab/8xh/QW3mohqBpQic8wA2ikNnn7k6H2bLEZDjMPHSel1zOhQH9sFzzoCbeJgVZ1//3FUGlLihGXVF795sWYclvHhPW2W4gtZYfCXJroKMydCI3AEAABECSURBVK6aw4UxAheFgMf4i2ouF9YILEfgnCsTf6jYfGOgs9dcfgQw9oh/LPfFzpovb5LTpRS+bC17Ibt2CPd0BVmR05Z1mMNLtBzK5kvS1Xd0LH5FlU+a1JicFG5nZgSuBgH1HR7jr6Y1XREjMIzAOZUJbpPhCs92uw5FVQcUygS38LTnAoizMQJGwAgYASNgBPaPgMZvj/H7byaX0AhsgsDZDmCro6nuM1et4rsE1RnfTWptJkbACBgBI2AEjMDREPAYfzRozdgI7A6Bc56ZOABDnQ83ErH1ho+apcPFB0QOMAJGwAgYASNgBC4OAY/xF9dkLrARmITArpQJlZgDwT+qw+n9YNqkWpnICBgBI2AEjIAR2BsCHuP31iIujxHYAIGznZkoyy4FIt0sJHvurUQlK/uNgBEwAkbACBiBHSHgMX5HjeGiGIGNEdjFyoQ6Gb4tcWdFYuPWNTsjYASMgBEwAmdGwGP8mRvA2RuBIyNwdmVCnUxcA9uuSCjsI54j193sjYARMAJGwAgYgSMi4DH+iOCatRHYCQJnVSbUyXBz03PZrSLR4IKCYWMEjIARMAJGwAhcKAIe4y+04VxsIzATgbOdmVAnw8pDfM05iv1Ujmd6PlF852N2QWDbCBgBI2AEjIAR2DcCHuP33T4unRHYEoGzfWdCleDjdCgU6bxEUSl/Y6IAxF4jYASMgBEwAheEgMf4C2osF9UIrEHgbCsTawrttOMIaFboW1Gx0oPChnL2D4X52x0C4hjGeB8DVfM0AkbACBgBI2AE9o7AOVcm9o7NxZavEWy/l51WeGQzQ/Sznr6vjl9sXfdQcOO9h1ZwGYyAETACRsAIGIFzIHDWA9jnqPCN5FkeYH+penMOhVUKm+0RMN7bY2qORsAIGAEjYASMwAUgYGXiAhppYRFzxSG2N+VhC9k6WQ8CObbGuwckBxsBI2AEjIARMALXhYC3OV1Xe6baaAWivAmLK3gxbx6sw99m1eKF7B8OY7shomEm/lfZPigvIISD8e6+IvYZASNgBIyAETACN4KAD2CfqaElgHI4mrMMn+p5WxFINyuZeP8mZpyh+K7GNMoi+6+1+FqYaCk7h7qtUBQACRPjXWBirxEwAkbACBgBI3CdCHib05naVQLnvR6Ed4Tx/x6rGMqDW53+K7uqSDT5cji7/HBgWySl/U1Pvo2HuC/1oFDYZAgY7wwMO42AETACRsAIGIGrR8DKxPmbmC1IfLxvcyPB9isxfSp7SFGA5o1oqisMCmdL00dlvPycC0BJqX0nRFG3Z4SF8b69ZneNjYARMAJGwAjcNAI+M3HG5pfw+aLJfvOViYb3x7KTItH42U71a1Hlf8g/tL2JuDJNsOCWqP/TM7TqEbRXbRvvq25eV84IGAEjYASMgBHoQcArEz3AnCg4bXOSIBq3/2ySrfix2sH2ptdyf8YjN0pDZ/VB4WnrkuxOuOhyg8JTVXaUjnL/T3YoRXm6m3Gr/sb7ZlrbFTUCRsAIGAEjYARyBLwykaNxendHUJdQitD/TA8fl/tW/lbIl5sD22yjwTzXg3KAMoAg+6Ge30UTKwScgYC+c6ZB8aXSQn4HioLoKNff9MA/5aGw7+Xm7ETkIW8ypIf2gM9D9Gl/m7KjpFFuMMJQTwy4oWB9pwfcvtDzexPOAfW2DnIH3tzU1Km34jhgTfvEzVc3i7dwsDECRsAIGAEjYARuGAErE2dq/EZYRaBlq9Cd/Jw9+FEPQjCC+y96krDa0H4jOwnHslEqOGeRbmiSH+EWRSEJ+vJ/IPcUg9DdUThIpPQI1ZyHQAjnuljo+gx5957J6Et0xPC/qrzcMpXwUT4/yR24oCD8IT/Zc7Vt4Ek9wSHHLeEtGhQr4kolCkyiffJ0Cu4114h3b2UdYQSMgBEwAkbACFw/Ao+vv4q7rSFCKiYOMf8gwZWVCL4F8bXcMesNzTd6ktKBR4bZcpSOoEGYZ3VgroHH24FECL995yUiGekR0s9uhBnK2eumINTtmcJyJeBZE/ehwttVCOj0tHVQHGmDD7iWGKGEtKtGck81V4X31EqbzggYASNgBIyAEbheBLwycb62RVBnNYHtMu3MvtyEhZIQpXvZhIcfoRklBNo72blgHDRTbIToxKOHGIWH1ZIhg1CNkLwHw/kNVhyeqjCUCYxzA26YXDHDD12OIQfVo96fV+jBLJQNOSeba8N7csVNaASMgBEwAkbACFwnAo+vs1oXUSsEdQRY9uOzFaf3ilXFlQI/aY9ynWwgpzxDID9qPpHfFnaGE/jcyZ8rCAShNKBwDOIZ8bLhAw6lcveJ4kLZUPR6I34Xh/f6WpuDETACRsAIGAEjcOkIWJk4QwtmgmM686AisDLBCkUSgoeKJJqYXS8F5aFkfXFDW5Q+JZHya/ORG4G3NMy2L9nyU/LZ0o/SUNueBb4dJUB1CoUhhRd17FM+OPC+xFwr3kuwcBojYASMgBEwAkbgChCwMnGeRmTrzJ0E1xDUY09+EtYVzrWurZGf611DkE8Kh/ytsCz337P4Nt0EB0oAykDNdARy8eeQMluHSkPYfRl4Zj/4BrapKA0+lLVcaeFMBFujWLEgPlfo8Lc4y30nGnD4F+4F5lrxXgCFkxgBI2AEjIARMALXgMDja6jEBdYBQb0VdiWghjB+MMOvOIRbbhMKIZe0rVE8Au99xqONm+BAUOYa2ppBeUnlEW/cfAW7I1g3iUjf1qUJO5vV4EF5S6UhlLCyrKzARBgH3/OVC+oPvsk0ODzvwSHIhuyrw3uoso4zAkbACBgBI2AErh+BJ5daxUZofCG73M9+UCXRMJucZp8PIs8TgLDL9a+54YYghFmuWs0PCL+RHwEXYZ5zFcykfyV3XB97J/coBkpTMwjcZTmCjtURtmGlb1vIzm9FChpshPEv84AzuxH+WWUIBSGKQ3gNJ7DmOlnq2cFCYVwx+x89YEG73BGGvdBcI94LoXAyI2AEjIARMAJG4BoQePTu3buLq4cEOoRxhLzOLP1QRUTL7D7C4cHs/1C6a48THgjJCNOzcVEa2uEX2VxVazMBAeM9ASSTGAEjYASMgBEwAheDwOOLKWm3oD/Ly6HlqkFg09NuT2mImD1HobDpIsCs+9LZ9m+UtjOb32VtXwUB410BxUFGwAgYASNgBIzAZSJwccqElAS2o7yRXZ1JVzhbmtgS1ImX/17h8YG4y2ytI5RauLD1B7xK5WswN9GzKsE2s77tT4Ppa5HixUHz3itya2kuLUz12w3el4ady2sEjIARMAJGwAjsD4GLUyYEIbPozO72GbY+1Q4KQ8/+eGbTbboIcA5j7goDqzyks5mPgPGej5lTGAEjYASMgBEwAjtE4KKUCc3qptlz2Z1VhwJXbu0pD98mEqVjdYLDuXEzUpH0Nr0NLhz+Toetx1Bo6KAfaocxNjcbb7xvtuldcSNgBIyAETACV4fAwW1OEnTYJsTsfrq9RjYCI4JjOuwsG0EcN4J97LUnDea5ntei+U4PH1f7Qs/vTTg3A3WEfPnZKhMCLGnhB1/S8mGw3+ElOwz5dHgQIRrKxGwvaVN6hTHTztmJPL2CUnpoD/gQeatGONHOtduODiAR7SS6g4QOaBEw3i0UdhgBI2AEjIARMAIXjEBHmZCAw/ahT2SH4oBgjlKRC4/c/MOtSHHN5U9yJ4FdNsrBH7JlpatYk7IhP0oA22I+IALT0H4jO2hQKhJPhaGMkC8rCbkyQLkODlGLFsWA8xDkwz7+VH65awa+vYe3awkcZgSMgBEwAkbACBgBI2AEjMAhAo8jqBHE4zsGKVhhsY0FIf9OflYMXqfIhxWAZwrLhf34mvKHCs9n/glH0cgNZxc4wxCG60VRXkJxQeBnBSE3xL/NAwo3SkTfeYkgJX1ZloizbQSMgBEwAkbACBgBI2AEjMBEBPKViVdKw+w+qwHJNMoD7lAMOG/Ax98QxhHsyxUAlA1MriTghy544Me8zPOSn7Rt/oor6UmDUtKWj4DCsN0p/4JxEZ28KEiUfdA0dfx5kOgw8kulG1RmFH95H/Y4rOdgiOr4aIhA8ayA0Val4b1CQWV7XGl490rlskOj+KvHtlPhxqN6D+JdS+MwI2AEjIARMAJGwAhsgUBSJiSMIMgjyJVbiBD4EOKSAB+2wpIgKH8p8KM0tPRyh4G+o2BkvHKatOUpAubY4kf5URLSKsqctDXapnx/rsWtCRPfmxf8hEG1nRXONjWuqc1XuybDbWwnQ2VCI2AEjIARMAJGwAhsgsDjhsunjV1TDsowSPu2E6E0dFYGJOARhqCfwuXH3TEKixWNWl457dAWpVQH8Wp51PISM1Y3YvtWzttuI2AEjIARMAJGwAgYASNgBGYgEMoEQvqdhO9SyEYRqM30f67wVmhv0vatDLA1ha1RrFiwcgBP8uIDZaFYRFi7RUhxf8/iSYKhfCgDNdNRcOAvIvIrDWH3ZaD9RsAIGAEjYASMgBEwAkbACMxDIJSJpBhIAG+Fb7ljy1OpNECDElAqGaEQdOhFx4pBhHHF7I96oIV/SiMbRaA1TTnuZZdCP8pG39YjypSUIaXDzXaZVjlpmT+kj/JkwXYaASNgBIyAETACRsAIGAEjMAeBJxAjtOtBoP9WNrc18Y0HBPPa+QeUCcJLgZzwuIlJztZwVoLrZLn6Nb6y/EZutj0h8McNUl/JTfwveu7krvFCgQkekOWGQ718yyJ9t0J23757lJsv84R2GwEjYASMgBEwAkbACBgBIzAfgUfv3tUvwJEwjuCO0sAVrbsxKg/fiUA5KbdkjZZRaVix+EU219BerWnqiXKFidUmvg1SW6l5oNrBr8qXtqbJ7lMEd1DK90W4VJzf18AuI2AEjIARMAJGwAisQ+AxySUUxQHoxK0RkmIr0roctk+NkFy9DWhCVnzbom9lY0Ly/ZM0bfdKNlvK4svlKBEoUbGtbK8VYbWrtiK1u/JeOM67w9MFMgJGwAgYASNgBC4TgccSihDOETSZtQ/DeYYfFFZuZYr4s9kqE8Im26Nixn1SWZr6vZB9EbPekypVJ6I9O9u4VGeUr3s9cQ6mnvLMoSpn7ZzMmUvVm/3F4txbI0cYASNgBIyAETACRmAmAqxMsJ2J8wufS5jjBiVm7jl7sKvtTSpTbrghau4KA4L04EfP8gwu2M1NW+ncSVEHFMOnatdZSljBw973CBjn91jYZQSMgBEwAkbACNwoAk8kXCJk7m4FYqg9VGZmsNnCw6Ht0W0x0Ikf9LPPWQyVY6dx1PET1RXFgdWI0uQrUGWc/dMRMM7TsTKlETACRsAIGAEjcKUI9B7AvtL63my1pFhwcJ1ViQ96lIybxWbLihvnLdE0LyNgBIyAETACRmDvCLDNyebKEZCAy8FrFInvrEgcr7GN8/GwNWcjYASMgBEwAkZgnwhYmdhnu2xdKs6X8LHApbdgbV2ea+VnnK+1ZV0vI2AEjIARMAJGoIqAtzlVYbmeQCkQ6aC67D0fqL94wI3zxTehK2AEjIARMAJGwAgsQMArEwtAu5QkEnD5uvidFYnjtphxPi6+5m4EjIARMAJGwAjsFwErE/ttm1Ulk4DL16Q/zhUJuWd/n2NVIW4gsXG+gUZ2FY2AETACRsAIGIFeBKxM9EJzuREScPmi+fNckWhqg4JhsxECxnkjIM3GCBgBI2AEjIARuFgEfGbiYpuuXnAJuB8pJj5EGER8W+KZHr4/8XEE2l6OgHFejp1TGgEjYASMgBEwAteDwJPrqYpr0iDAl75RKNJ5iQKVW/hoX1Hlo3mN89GgNWMjYASMgBEwAkbgUhD4f/B+Pbau3XTHAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\frac{I_{c} \\omega^{2}{\\left(t \\right)}}{2} + M g \\left(l \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + x{\\left(t \\right)}\\right) + \\frac{M \\left(l^{2} \\omega^{2}{\\left(t \\right)} \\cos^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(- l \\omega{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + v{\\left(t \\right)}\\right)^{2}\\right)}{2} + g m x{\\left(t \\right)} - \\frac{k x^{2}{\\left(t \\right)}}{2} + \\frac{m v^{2}{\\left(t \\right)}}{2}$"
      ],
      "text/plain": [
       "     2                                   ⎛ 2  2       2                       \n",
       "I_c⋅ω (t)                              M⋅⎝l ⋅ω (t)⋅cos (θ(t)) + (-l⋅ω(t)⋅sin(θ\n",
       "───────── + M⋅g⋅(l⋅cos(θ(t)) + x(t)) + ───────────────────────────────────────\n",
       "    2                                                            2            \n",
       "\n",
       "            2⎞                 2         2   \n",
       "(t)) + v(t)) ⎠              k⋅x (t)   m⋅v (t)\n",
       "────────────── + g⋅m⋅x(t) - ─────── + ───────\n",
       "                               2         2   "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = T - U\n",
    "L"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - M g + \\frac{M \\left(- 2 l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - 2 l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\omega{\\left(t \\right)} + 2 \\frac{d}{d t} v{\\left(t \\right)}\\right)}{2} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)} + m \\frac{d}{d t} v{\\left(t \\right)}$"
      ],
      "text/plain": [
       "         ⎛       2                              d            d       ⎞        \n",
       "       M⋅⎜- 2⋅l⋅ω (t)⋅cos(θ(t)) - 2⋅l⋅sin(θ(t))⋅──(ω(t)) + 2⋅──(v(t))⎟        \n",
       "         ⎝                                      dt           dt      ⎠        \n",
       "-M⋅g + ─────────────────────────────────────────────────────────────── + b⋅v(t\n",
       "                                      2                                       \n",
       "\n",
       "                             \n",
       "                             \n",
       "                     d       \n",
       ") - g⋅m + k⋅x(t) + m⋅──(v(t))\n",
       "                     dt      "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "leq_v = L.diff(v).diff(t).subs(eq_repl) - L.diff(x) -(-R.diff(v))\n",
    "leq_v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle I_{c} \\frac{d}{d t} \\omega{\\left(t \\right)} + M g l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} - \\frac{M \\left(- 2 l^{2} \\omega^{2}{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - 2 l \\left(- l \\omega{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + v{\\left(t \\right)}\\right) \\omega{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)}\\right)}{2} + \\frac{M \\left(- 4 l^{2} \\omega^{2}{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + 2 l^{2} \\cos^{2}{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\omega{\\left(t \\right)} - 2 l \\left(- l \\omega{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + v{\\left(t \\right)}\\right) \\omega{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - 2 l \\left(- l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\omega{\\left(t \\right)} + \\frac{d}{d t} v{\\left(t \\right)}\\right) \\sin{\\left(\\theta{\\left(t \\right)} \\right)}\\right)}{2}$"
      ],
      "text/plain": [
       "                                                                              \n",
       "                                   ⎛     2  2                                 \n",
       "    d                            M⋅⎝- 2⋅l ⋅ω (t)⋅sin(θ(t))⋅cos(θ(t)) - 2⋅l⋅(-l\n",
       "I_c⋅──(ω(t)) + M⋅g⋅l⋅sin(θ(t)) - ─────────────────────────────────────────────\n",
       "    dt                                                                    2   \n",
       "\n",
       "                                            ⎛     2  2                        \n",
       "                                      ⎞   M⋅⎜- 4⋅l ⋅ω (t)⋅sin(θ(t))⋅cos(θ(t)) \n",
       "⋅ω(t)⋅sin(θ(t)) + v(t))⋅ω(t)⋅cos(θ(t))⎠     ⎝                                 \n",
       "─────────────────────────────────────── + ────────────────────────────────────\n",
       "                                                                              \n",
       "\n",
       "     2    2       d                                                           \n",
       "+ 2⋅l ⋅cos (θ(t))⋅──(ω(t)) - 2⋅l⋅(-l⋅ω(t)⋅sin(θ(t)) + v(t))⋅ω(t)⋅cos(θ(t)) - 2\n",
       "                  dt                                                          \n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                       2                      \n",
       "\n",
       "   ⎛     2                            d          d       ⎞          ⎞\n",
       "⋅l⋅⎜- l⋅ω (t)⋅cos(θ(t)) - l⋅sin(θ(t))⋅──(ω(t)) + ──(v(t))⎟⋅sin(θ(t))⎟\n",
       "   ⎝                                  dt         dt      ⎠          ⎠\n",
       "─────────────────────────────────────────────────────────────────────\n",
       "                                                                     "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "leq_omega = L.diff(omega).diff(t).subs(eq_repl) - L.diff(theta) -(-R.diff(omega))\n",
    "leq_omega"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle I_{c} \\frac{d}{d t} \\omega{\\left(t \\right)} + M g l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + M l^{2} \\frac{d}{d t} \\omega{\\left(t \\right)} - M l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} v{\\left(t \\right)}$"
      ],
      "text/plain": [
       "    d                               2 d                        d       \n",
       "I_c⋅──(ω(t)) + M⋅g⋅l⋅sin(θ(t)) + M⋅l ⋅──(ω(t)) - M⋅l⋅sin(θ(t))⋅──(v(t))\n",
       "    dt                                dt                       dt      "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "leq_omega = sm.simplify(leq_omega)\n",
    "leq_omega"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Put EoM in explicit first order form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- M g + \\frac{M \\left(- 2 l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - 2 l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\omega{\\left(t \\right)} + 2 \\frac{d}{d t} v{\\left(t \\right)}\\right)}{2} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)} + m \\frac{d}{d t} v{\\left(t \\right)}\\\\I_{c} \\frac{d}{d t} \\omega{\\left(t \\right)} + M g l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + M l^{2} \\frac{d}{d t} \\omega{\\left(t \\right)} - M l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} v{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡         ⎛       2                              d            d       ⎞       \n",
       "⎢       M⋅⎜- 2⋅l⋅ω (t)⋅cos(θ(t)) - 2⋅l⋅sin(θ(t))⋅──(ω(t)) + 2⋅──(v(t))⎟       \n",
       "⎢         ⎝                                      dt           dt      ⎠       \n",
       "⎢-M⋅g + ─────────────────────────────────────────────────────────────── + b⋅v(\n",
       "⎢                                      2                                      \n",
       "⎢                                                                             \n",
       "⎢                      d                               2 d                    \n",
       "⎢                  I_c⋅──(ω(t)) + M⋅g⋅l⋅sin(θ(t)) + M⋅l ⋅──(ω(t)) - M⋅l⋅sin(θ(\n",
       "⎣                      dt                                dt                   \n",
       "\n",
       "                              ⎤\n",
       "                              ⎥\n",
       "                      d       ⎥\n",
       "t) - g⋅m + k⋅x(t) + m⋅──(v(t))⎥\n",
       "                      dt      ⎥\n",
       "                              ⎥\n",
       "    d                         ⎥\n",
       "t))⋅──(v(t))                  ⎥\n",
       "    dt                        ⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sm.Matrix([leq_v, leq_omega])\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}\\\\M g l \\sin{\\left(\\theta{\\left(t \\right)} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡            2                                     ⎤\n",
       "⎢-M⋅g - M⋅l⋅ω (t)⋅cos(θ(t)) + b⋅v(t) - g⋅m + k⋅x(t)⎥\n",
       "⎢                                                  ⎥\n",
       "⎣                 M⋅g⋅l⋅sin(θ(t))                  ⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = f.subs({omega.diff(t): 0, v.diff(t): 0})\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- M l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} & M + m\\\\I_{c} + M l^{2} & - M l \\sin{\\left(\\theta{\\left(t \\right)} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-M⋅l⋅sin(θ(t))      M + m     ⎤\n",
       "⎢                              ⎥\n",
       "⎢           2                  ⎥\n",
       "⎣  I_c + M⋅l     -M⋅l⋅sin(θ(t))⎦"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I = f.jacobian([omega.diff(t), v.diff(t)])\n",
    "I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{M g l \\left(M + m\\right) \\sin{\\left(\\theta{\\left(t \\right)} \\right)}}{- M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(I_{c} + M l^{2}\\right) \\left(M + m\\right)} - \\frac{M l \\left(- M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}\\right) \\sin{\\left(\\theta{\\left(t \\right)} \\right)}}{- M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(I_{c} + M l^{2}\\right) \\left(M + m\\right)}\\\\- \\frac{M^{2} g l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{- M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(I_{c} + M l^{2}\\right) \\left(M + m\\right)} - \\frac{\\left(I_{c} + M l^{2}\\right) \\left(- M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}\\right)}{- M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(I_{c} + M l^{2}\\right) \\left(M + m\\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                                  ⎛            2             \n",
       "⎢           M⋅g⋅l⋅(M + m)⋅sin(θ(t))            M⋅l⋅⎝-M⋅g - M⋅l⋅ω (t)⋅cos(θ(t))\n",
       "⎢- ───────────────────────────────────────── - ───────────────────────────────\n",
       "⎢     2  2    2         ⎛         2⎞                          2  2    2       \n",
       "⎢  - M ⋅l ⋅sin (θ(t)) + ⎝I_c + M⋅l ⎠⋅(M + m)               - M ⋅l ⋅sin (θ(t)) \n",
       "⎢                                                                             \n",
       "⎢               2    2    2                    ⎛         2⎞ ⎛            2    \n",
       "⎢              M ⋅g⋅l ⋅sin (θ(t))              ⎝I_c + M⋅l ⎠⋅⎝-M⋅g - M⋅l⋅ω (t)⋅\n",
       "⎢- ───────────────────────────────────────── - ───────────────────────────────\n",
       "⎢     2  2    2         ⎛         2⎞                          2  2    2       \n",
       "⎣  - M ⋅l ⋅sin (θ(t)) + ⎝I_c + M⋅l ⎠⋅(M + m)               - M ⋅l ⋅sin (θ(t)) \n",
       "\n",
       "                        ⎞          ⎤\n",
       " + b⋅v(t) - g⋅m + k⋅x(t)⎠⋅sin(θ(t))⎥\n",
       "───────────────────────────────────⎥\n",
       "  ⎛         2⎞                     ⎥\n",
       "+ ⎝I_c + M⋅l ⎠⋅(M + m)             ⎥\n",
       "                                   ⎥\n",
       "                                 ⎞ ⎥\n",
       "cos(θ(t)) + b⋅v(t) - g⋅m + k⋅x(t)⎠ ⎥\n",
       "────────────────────────────────── ⎥\n",
       "  ⎛         2⎞                     ⎥\n",
       "+ ⎝I_c + M⋅l ⎠⋅(M + m)             ⎦"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sdot = -I.inv()*g\n",
    "sdot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{M g l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + \\frac{M l \\left(\\frac{M^{2} g l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} - M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}\\right) \\sin{\\left(\\theta{\\left(t \\right)} \\right)}}{- \\frac{M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} + M + m}}{I_{c} + M l^{2}}\\\\- \\frac{\\frac{M^{2} g l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} - M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}}{- \\frac{M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} + M + m}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ ⎛                      ⎛ 2    2    2                                        \n",
       "⎢ ⎜                      ⎜M ⋅g⋅l ⋅sin (θ(t))              2                   \n",
       "⎢ ⎜                  M⋅l⋅⎜────────────────── - M⋅g - M⋅l⋅ω (t)⋅cos(θ(t)) + b⋅v\n",
       "⎢ ⎜                      ⎜             2                                      \n",
       "⎢ ⎜                      ⎝    I_c + M⋅l                                       \n",
       "⎢-⎜M⋅g⋅l⋅sin(θ(t)) + ─────────────────────────────────────────────────────────\n",
       "⎢ ⎜                                                   2  2    2               \n",
       "⎢ ⎜                                                  M ⋅l ⋅sin (θ(t))         \n",
       "⎢ ⎜                                                - ──────────────── + M + m \n",
       "⎢ ⎜                                                              2            \n",
       "⎢ ⎝                                                     I_c + M⋅l             \n",
       "⎢─────────────────────────────────────────────────────────────────────────────\n",
       "⎢                                                          2                  \n",
       "⎢                                                 I_c + M⋅l                   \n",
       "⎢                                                                             \n",
       "⎢                  ⎛ 2    2    2                                              \n",
       "⎢                  ⎜M ⋅g⋅l ⋅sin (θ(t))              2                         \n",
       "⎢                 -⎜────────────────── - M⋅g - M⋅l⋅ω (t)⋅cos(θ(t)) + b⋅v(t) - \n",
       "⎢                  ⎜             2                                            \n",
       "⎢                  ⎝    I_c + M⋅l                                             \n",
       "⎢                 ────────────────────────────────────────────────────────────\n",
       "⎢                                            2  2    2                        \n",
       "⎢                                           M ⋅l ⋅sin (θ(t))                  \n",
       "⎢                                         - ──────────────── + M + m          \n",
       "⎢                                                       2                     \n",
       "⎣                                              I_c + M⋅l                      \n",
       "\n",
       "                  ⎞          ⎞ ⎤\n",
       "                  ⎟          ⎟ ⎥\n",
       "(t) - g⋅m + k⋅x(t)⎟⋅sin(θ(t))⎟ ⎥\n",
       "                  ⎟          ⎟ ⎥\n",
       "                  ⎠          ⎟ ⎥\n",
       "─────────────────────────────⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎠ ⎥\n",
       "───────────────────────────────⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "            ⎞                  ⎥\n",
       "            ⎟                  ⎥\n",
       "g⋅m + k⋅x(t)⎟                  ⎥\n",
       "            ⎟                  ⎥\n",
       "            ⎠                  ⎥\n",
       "──────────────                 ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎦"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sdot = -I.LUsolve(g)\n",
    "sdot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{M l \\left(- M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} + k x{\\left(t \\right)}\\right) \\sin{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} M + I_{c} m + M^{2} l^{2} \\cos^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + M l^{2} m}\\\\\\frac{- M^{2} g l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(- I_{c} - M l^{2}\\right) \\left(- M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}\\right)}{- M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(I_{c} + M l^{2}\\right) \\left(M + m\\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                     ⎛       2                               ⎞               \n",
       "⎢                -M⋅l⋅⎝- M⋅l⋅ω (t)⋅cos(θ(t)) + b⋅v(t) + k⋅x(t)⎠⋅sin(θ(t))     \n",
       "⎢                ─────────────────────────────────────────────────────────    \n",
       "⎢                                         2  2    2            2              \n",
       "⎢                        I_c⋅M + I_c⋅m + M ⋅l ⋅cos (θ(t)) + M⋅l ⋅m            \n",
       "⎢                                                                             \n",
       "⎢   2    2    2         ⎛          2⎞ ⎛            2                          \n",
       "⎢- M ⋅g⋅l ⋅sin (θ(t)) + ⎝-I_c - M⋅l ⎠⋅⎝-M⋅g - M⋅l⋅ω (t)⋅cos(θ(t)) + b⋅v(t) - g\n",
       "⎢─────────────────────────────────────────────────────────────────────────────\n",
       "⎢                           2  2    2         ⎛         2⎞                    \n",
       "⎣                        - M ⋅l ⋅sin (θ(t)) + ⎝I_c + M⋅l ⎠⋅(M + m)            \n",
       "\n",
       "            ⎤\n",
       "            ⎥\n",
       "            ⎥\n",
       "            ⎥\n",
       "            ⎥\n",
       "            ⎥\n",
       "           ⎞⎥\n",
       "⋅m + k⋅x(t)⎠⎥\n",
       "────────────⎥\n",
       "            ⎥\n",
       "            ⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.simplify(sdot)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulate the non-linear system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "from resonance.nonlinear_systems import MultiDoFNonLinearSystem"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys = MultiDoFNonLinearSystem()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys.constants['m'] = 1.0  # kg\n",
    "sys.constants['k'] = 10.0  # N/m\n",
    "sys.constants['b'] = 5.0  # Ns\n",
    "sys.constants['l'] = 0.5  # m\n",
    "sys.constants['M'] = 0.5  # kg\n",
    "sys.constants['Ic'] = 0.5*0.5**2  # kg m**2\n",
    "sys.constants['g'] = 9.81  # m/s**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# order of entry matters!\n",
    "sys.coordinates['theta'] = 1.0  # rad\n",
    "sys.coordinates['x'] = 0.0  # m\n",
    "sys.speeds['omega'] = 0.0 # rad/s\n",
    "sys.speeds['v'] = 0.0 # m/s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "_StatesDict([('theta', 1.0), ('x', 0.0), ('omega', 0.0), ('v', 0.0)])"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys.states"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "eval_mass = sm.lambdify((m, M), m+M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3.0$"
      ],
      "text/plain": [
       "3.0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eval_mass(1.0, 2.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = sm.symbols('g', real=True, positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "eval_sdot = sm.lambdify((theta, x, omega, v, m, k, b, l, M, Ic, g), [sdot[0], sdot[1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 4.144543155486223, \\  51.08691546911525\\right]$"
      ],
      "text/plain": [
       "[4.144543155486223, 51.08691546911525]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eval_sdot(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def eval_derivatives(theta, x, omega, v, m, k, b, l, M, Ic, g):\n",
    "    omegadot, vdot = eval_sdot(theta, x, omega, v, m, k, b, l, M, Ic, g)\n",
    "    thetadot = omega\n",
    "    xdot = v\n",
    "    return thetadot, xdot, omegadot, vdot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 3.0, \\  4.0, \\  4.144543155486223, \\  51.08691546911525\\right)$"
      ],
      "text/plain": [
       "(3.0, 4.0, 4.144543155486223, 51.08691546911525)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eval_derivatives(1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys.diff_eq_func = eval_derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "trajectories = sys.free_response(5.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib widget"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "a655cf3a4562408d888d84b4c290b7ee",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f8dab65f4a8>,\n",
       "       <matplotlib.axes._subplots.AxesSubplot object at 0x7f8dab5d3be0>],\n",
       "      dtype=object)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trajectories[['x', 'theta']].plot(subplots=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Find the equilibrium"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{M g l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + \\frac{M l \\left(\\frac{M^{2} g l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} - M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}\\right) \\sin{\\left(\\theta{\\left(t \\right)} \\right)}}{- \\frac{M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} + M + m}}{I_{c} + M l^{2}}\\\\- \\frac{\\frac{M^{2} g l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} - M g - M l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)}}{- \\frac{M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} + M l^{2}} + M + m}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ ⎛                      ⎛ 2    2    2                                        \n",
       "⎢ ⎜                      ⎜M ⋅g⋅l ⋅sin (θ(t))              2                   \n",
       "⎢ ⎜                  M⋅l⋅⎜────────────────── - M⋅g - M⋅l⋅ω (t)⋅cos(θ(t)) + b⋅v\n",
       "⎢ ⎜                      ⎜             2                                      \n",
       "⎢ ⎜                      ⎝    I_c + M⋅l                                       \n",
       "⎢-⎜M⋅g⋅l⋅sin(θ(t)) + ─────────────────────────────────────────────────────────\n",
       "⎢ ⎜                                                   2  2    2               \n",
       "⎢ ⎜                                                  M ⋅l ⋅sin (θ(t))         \n",
       "⎢ ⎜                                                - ──────────────── + M + m \n",
       "⎢ ⎜                                                              2            \n",
       "⎢ ⎝                                                     I_c + M⋅l             \n",
       "⎢─────────────────────────────────────────────────────────────────────────────\n",
       "⎢                                                          2                  \n",
       "⎢                                                 I_c + M⋅l                   \n",
       "⎢                                                                             \n",
       "⎢                  ⎛ 2    2    2                                              \n",
       "⎢                  ⎜M ⋅g⋅l ⋅sin (θ(t))              2                         \n",
       "⎢                 -⎜────────────────── - M⋅g - M⋅l⋅ω (t)⋅cos(θ(t)) + b⋅v(t) - \n",
       "⎢                  ⎜             2                                            \n",
       "⎢                  ⎝    I_c + M⋅l                                             \n",
       "⎢                 ────────────────────────────────────────────────────────────\n",
       "⎢                                            2  2    2                        \n",
       "⎢                                           M ⋅l ⋅sin (θ(t))                  \n",
       "⎢                                         - ──────────────── + M + m          \n",
       "⎢                                                       2                     \n",
       "⎣                                              I_c + M⋅l                      \n",
       "\n",
       "                  ⎞          ⎞ ⎤\n",
       "                  ⎟          ⎟ ⎥\n",
       "(t) - g⋅m + k⋅x(t)⎟⋅sin(θ(t))⎟ ⎥\n",
       "                  ⎟          ⎟ ⎥\n",
       "                  ⎠          ⎟ ⎥\n",
       "─────────────────────────────⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎟ ⎥\n",
       "                             ⎠ ⎥\n",
       "───────────────────────────────⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "            ⎞                  ⎥\n",
       "            ⎟                  ⎥\n",
       "g⋅m + k⋅x(t)⎟                  ⎥\n",
       "            ⎟                  ⎥\n",
       "            ⎠                  ⎥\n",
       "──────────────                 ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎥\n",
       "                               ⎦"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sdot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "sdot_no_motion = sdot.subs({omega: 0, v: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{M k l x{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)}}{I_{c} M + I_{c} m + M^{2} l^{2} \\cos^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + M l^{2} m}\\\\\\frac{- M^{2} g l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} - \\left(I_{c} + M l^{2}\\right) \\left(- M g - g m + k x{\\left(t \\right)}\\right)}{- M^{2} l^{2} \\sin^{2}{\\left(\\theta{\\left(t \\right)} \\right)} + \\left(I_{c} + M l^{2}\\right) \\left(M + m\\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                  -M⋅k⋅l⋅x(t)⋅sin(θ(t))                  ⎤\n",
       "⎢        ─────────────────────────────────────────        ⎥\n",
       "⎢                         2  2    2            2          ⎥\n",
       "⎢        I_c⋅M + I_c⋅m + M ⋅l ⋅cos (θ(t)) + M⋅l ⋅m        ⎥\n",
       "⎢                                                         ⎥\n",
       "⎢   2    2    2         ⎛         2⎞                      ⎥\n",
       "⎢- M ⋅g⋅l ⋅sin (θ(t)) - ⎝I_c + M⋅l ⎠⋅(-M⋅g - g⋅m + k⋅x(t))⎥\n",
       "⎢─────────────────────────────────────────────────────────⎥\n",
       "⎢           2  2    2         ⎛         2⎞                ⎥\n",
       "⎣        - M ⋅l ⋅sin (θ(t)) + ⎝I_c + M⋅l ⎠⋅(M + m)        ⎦"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.simplify(sdot_no_motion)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ \\theta{\\left(t \\right)} : 0, \\  x{\\left(t \\right)} : \\frac{g \\left(M + m\\right)}{k}\\right\\}, \\  \\left\\{ \\theta{\\left(t \\right)} : 0, \\  x{\\left(t \\right)} : \\frac{g \\left(M + m\\right)}{k}\\right\\}, \\  \\left\\{ \\theta{\\left(t \\right)} : \\pi, \\  x{\\left(t \\right)} : \\frac{g \\left(M + m\\right)}{k}\\right\\}, \\  \\left\\{ \\theta{\\left(t \\right)} : \\pi, \\  x{\\left(t \\right)} : \\frac{g \\left(M + m\\right)}{k}\\right\\}\\right]$"
      ],
      "text/plain": [
       "⎡⎧               g⋅(M + m)⎫  ⎧               g⋅(M + m)⎫  ⎧               g⋅(M \n",
       "⎢⎨θ(t): 0, x(t): ─────────⎬, ⎨θ(t): 0, x(t): ─────────⎬, ⎨θ(t): π, x(t): ─────\n",
       "⎣⎩                   k    ⎭  ⎩                   k    ⎭  ⎩                   k\n",
       "\n",
       "+ m)⎫  ⎧               g⋅(M + m)⎫⎤\n",
       "────⎬, ⎨θ(t): π, x(t): ─────────⎬⎥\n",
       "    ⎭  ⎩                   k    ⎭⎦"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sm.solve(sdot_no_motion, theta, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linearize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- M g + \\frac{M \\left(- 2 l \\omega^{2}{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - 2 l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\omega{\\left(t \\right)} + 2 \\frac{d}{d t} v{\\left(t \\right)}\\right)}{2} + b v{\\left(t \\right)} - g m + k x{\\left(t \\right)} + m \\frac{d}{d t} v{\\left(t \\right)}\\\\I_{c} \\frac{d}{d t} \\omega{\\left(t \\right)} + M g l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} + M l^{2} \\frac{d}{d t} \\omega{\\left(t \\right)} - M l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} v{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡         ⎛       2                              d            d       ⎞       \n",
       "⎢       M⋅⎜- 2⋅l⋅ω (t)⋅cos(θ(t)) - 2⋅l⋅sin(θ(t))⋅──(ω(t)) + 2⋅──(v(t))⎟       \n",
       "⎢         ⎝                                      dt           dt      ⎠       \n",
       "⎢-M⋅g + ─────────────────────────────────────────────────────────────── + b⋅v(\n",
       "⎢                                      2                                      \n",
       "⎢                                                                             \n",
       "⎢                      d                               2 d                    \n",
       "⎢                  I_c⋅──(ω(t)) + M⋅g⋅l⋅sin(θ(t)) + M⋅l ⋅──(ω(t)) - M⋅l⋅sin(θ(\n",
       "⎣                      dt                                dt                   \n",
       "\n",
       "                              ⎤\n",
       "                              ⎥\n",
       "                      d       ⎥\n",
       "t) - g⋅m + k⋅x(t) + m⋅──(v(t))⎥\n",
       "                      dt      ⎥\n",
       "                              ⎥\n",
       "    d                         ⎥\n",
       "t))⋅──(v(t))                  ⎥\n",
       "    dt                        ⎦"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{d}{d t} \\omega{\\left(t \\right)}\\\\\\frac{d}{d t} v{\\left(t \\right)}\\\\\\omega{\\left(t \\right)}\\\\v{\\left(t \\right)}\\\\\\theta{\\left(t \\right)}\\\\x{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡d       ⎤\n",
       "⎢──(ω(t))⎥\n",
       "⎢dt      ⎥\n",
       "⎢        ⎥\n",
       "⎢d       ⎥\n",
       "⎢──(v(t))⎥\n",
       "⎢dt      ⎥\n",
       "⎢        ⎥\n",
       "⎢  ω(t)  ⎥\n",
       "⎢        ⎥\n",
       "⎢  v(t)  ⎥\n",
       "⎢        ⎥\n",
       "⎢  θ(t)  ⎥\n",
       "⎢        ⎥\n",
       "⎣  x(t)  ⎦"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vbar = sm.Matrix([omega.diff(t), v.diff(t), omega, v, theta, x])\n",
    "vbar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0\\\\0\\\\0\\\\0\\\\0\\\\\\frac{g \\left(M + m\\right)}{k}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    0    ⎤\n",
       "⎢         ⎥\n",
       "⎢    0    ⎥\n",
       "⎢         ⎥\n",
       "⎢    0    ⎥\n",
       "⎢         ⎥\n",
       "⎢    0    ⎥\n",
       "⎢         ⎥\n",
       "⎢    0    ⎥\n",
       "⎢         ⎥\n",
       "⎢g⋅(M + m)⎥\n",
       "⎢─────────⎥\n",
       "⎣    k    ⎦"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vbareq = sm.Matrix([0, 0, 0, 0, 0, g*(M + m)/k])\n",
    "vbareq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\omega{\\left(t \\right)} : 0, \\  \\theta{\\left(t \\right)} : 0, \\  v{\\left(t \\right)} : 0, \\  x{\\left(t \\right)} : \\frac{g \\left(M + m\\right)}{k}, \\  \\frac{d}{d t} \\omega{\\left(t \\right)} : 0, \\  \\frac{d}{d t} v{\\left(t \\right)} : 0\\right\\}$"
      ],
      "text/plain": [
       "⎧                                 g⋅(M + m)  d            d          ⎫\n",
       "⎨ω(t): 0, θ(t): 0, v(t): 0, x(t): ─────────, ──(ω(t)): 0, ──(v(t)): 0⎬\n",
       "⎩                                     k      dt           dt         ⎭"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vbar_repl = {k: v for k, v in zip(vbar, vbareq)}  # dictionary comphrension\n",
    "vbar_repl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- M g + b v{\\left(t \\right)} - g m + g \\left(M + m\\right) + k \\left(- \\frac{g \\left(M + m\\right)}{k} + x{\\left(t \\right)}\\right) + \\left(M + m\\right) \\frac{d}{d t} v{\\left(t \\right)}\\\\M g l \\theta{\\left(t \\right)} + \\left(I_{c} + M l^{2}\\right) \\frac{d}{d t} \\omega{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                    ⎛  g⋅(M + m)       ⎞           d       ⎤\n",
       "⎢-M⋅g + b⋅v(t) - g⋅m + g⋅(M + m) + k⋅⎜- ───────── + x(t)⎟ + (M + m)⋅──(v(t))⎥\n",
       "⎢                                    ⎝      k           ⎠           dt      ⎥\n",
       "⎢                                                                           ⎥\n",
       "⎢                                 ⎛         2⎞ d                            ⎥\n",
       "⎢                    M⋅g⋅l⋅θ(t) + ⎝I_c + M⋅l ⎠⋅──(ω(t))                     ⎥\n",
       "⎣                                              dt                           ⎦"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "flin = f.subs(vbar_repl) + f.jacobian(vbar).subs(vbar_repl)*(vbar - vbareq)\n",
    "flin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- M l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} & M + m & - 2 M l \\omega{\\left(t \\right)} \\cos{\\left(\\theta{\\left(t \\right)} \\right)} & b & \\frac{M \\left(2 l \\omega^{2}{\\left(t \\right)} \\sin{\\left(\\theta{\\left(t \\right)} \\right)} - 2 l \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} \\omega{\\left(t \\right)}\\right)}{2} & k\\\\I_{c} + M l^{2} & - M l \\sin{\\left(\\theta{\\left(t \\right)} \\right)} & 0 & 0 & M g l \\cos{\\left(\\theta{\\left(t \\right)} \\right)} - M l \\cos{\\left(\\theta{\\left(t \\right)} \\right)} \\frac{d}{d t} v{\\left(t \\right)} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                                            ⎛     2          \n",
       "⎢                                                          M⋅⎜2⋅l⋅ω (t)⋅sin(θ(\n",
       "⎢                                                            ⎝                \n",
       "⎢-M⋅l⋅sin(θ(t))      M + m       -2⋅M⋅l⋅ω(t)⋅cos(θ(t))  b  ───────────────────\n",
       "⎢                                                                             \n",
       "⎢                                                                             \n",
       "⎢           2                                                                 \n",
       "⎢  I_c + M⋅l     -M⋅l⋅sin(θ(t))            0            0      M⋅g⋅l⋅cos(θ(t))\n",
       "⎣                                                                             \n",
       "\n",
       "                    d       ⎞   ⎤\n",
       "t)) - 2⋅l⋅cos(θ(t))⋅──(ω(t))⎟   ⎥\n",
       "                    dt      ⎠   ⎥\n",
       "─────────────────────────────  k⎥\n",
       "    2                           ⎥\n",
       "                                ⎥\n",
       "                 d              ⎥\n",
       " - M⋅l⋅cos(θ(t))⋅──(v(t))      0⎥\n",
       "                 dt             ⎦"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.jacobian(vbar)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Linear canonical form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\theta{\\left(t \\right)}\\\\x{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡θ(t)⎤\n",
       "⎢    ⎥\n",
       "⎣x(t)⎦"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cbar = sm.Matrix([theta, x])\n",
    "cbar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & k\\\\M g l & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡  0    k⎤\n",
       "⎢        ⎥\n",
       "⎣M⋅g⋅l  0⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = flin.jacobian(cbar)\n",
    "K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\omega{\\left(t \\right)}\\\\v{\\left(t \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ω(t)⎤\n",
       "⎢    ⎥\n",
       "⎣v(t)⎦"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sbar = sm.Matrix([omega, v])\n",
    "sbar"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & b\\\\0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0  b⎤\n",
       "⎢    ⎥\n",
       "⎣0  0⎦"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C = flin.jacobian(sbar)\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & M + m\\\\I_{c} + M l^{2} & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    0       M + m⎤\n",
       "⎢                 ⎥\n",
       "⎢         2       ⎥\n",
       "⎣I_c + M⋅l     0  ⎦"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M_mat = flin.jacobian(sbar.diff(t))\n",
    "M_mat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\omega{\\left(t \\right)} : 0, \\  \\theta{\\left(t \\right)} : 0, \\  v{\\left(t \\right)} : 0, \\  x{\\left(t \\right)} : 0, \\  \\frac{d}{d t} \\omega{\\left(t \\right)} : 0, \\  \\frac{d}{d t} v{\\left(t \\right)} : 0\\right\\}$"
      ],
      "text/plain": [
       "⎧                                    d            d          ⎫\n",
       "⎨ω(t): 0, θ(t): 0, v(t): 0, x(t): 0, ──(ω(t)): 0, ──(v(t)): 0⎬\n",
       "⎩                                    dt           dt         ⎭"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zero_repl = {k: v for k, v in zip(vbar, [0, 0, 0, 0, 0, 0])}\n",
    "zero_repl\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}M g + g m\\\\0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡M⋅g + g⋅m⎤\n",
       "⎢         ⎥\n",
       "⎣    0    ⎦"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = -flin.subs(zero_repl)\n",
    "F"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Simulate the linear system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "from resonance.linear_systems import MultiDoFLinearSystem\n",
    "\n",
    "sys = MultiDoFLinearSystem()\n",
    "\n",
    "sys.constants['m'] = 1.0  # kg\n",
    "sys.constants['k'] = 10.0  # N/m\n",
    "sys.constants['b'] = 5.0  # Ns\n",
    "sys.constants['l'] = 0.5  # m\n",
    "sys.constants['M'] = 0.5  # kg\n",
    "sys.constants['Ic'] = 0.5*0.5**2  # kg m**2\n",
    "sys.constants['g'] = 9.81  # m/s**2\n",
    "sys.coordinates['theta'] = 1.0  # rad\n",
    "sys.coordinates['x'] = 0.0  # m\n",
    "sys.speeds['omega'] = 0.0 # rad/s\n",
    "sys.speeds['v'] = 0.0 # m/s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 2],\n",
       "       [3, 4]])"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.array([[1, 2], [3, 4]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "def calc_coeff_mats(m, k, b, l, M, Ic, g):\n",
    "    M_mat = np.array([[0, M+m],\n",
    "                  [Ic+m*l**2, 0]])\n",
    "    C = np.array([[0, b],\n",
    "                  [0, 0]])\n",
    "    K = np.array([[0, k],\n",
    "                  [M*g*l, 0]])\n",
    "    return M_mat, C, K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys.canonical_coeffs_func = calc_coeff_mats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[0.   , 1.5  ],\n",
       "        [0.375, 0.   ]]), array([[0., 5.],\n",
       "        [0., 0.]]), array([[ 0.    , 10.    ],\n",
       "        [ 2.4525,  0.    ]]))"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys.canonical_coefficients()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys.coordinates['x'] = 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "traj = sys.free_response(5.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "3c4768da2ffb47d694e933616184cf86",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f8dab554f60>,\n",
       "       <matplotlib.axes._subplots.AxesSubplot object at 0x7f8dab444ba8>],\n",
       "      dtype=object)"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "traj[['x', 'theta']].plot(subplots=True)"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}