{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "39be5d38",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import init_printing, symbols, diff, simplify, integrate, lambdify, Rational, solveset, Eq, \\\n",
    "    cancel, Rational, factor\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "from IPython.display import display\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a28abe4c",
   "metadata": {},
   "outputs": [],
   "source": [
    "x,y, m,t, H, Ha, H0, H1, g, T, k, I, Ta, l, a = symbols(\"x y m t H H_alpha H_0 H_1 g T k I T_alpha l a\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "de84f91c",
   "metadata": {},
   "source": [
    "Let's now look at the natural tragjectory of an object thrown in to the air."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0d6d1c7",
   "metadata": {},
   "source": [
    "This can also be written as $ x = H_0 - \\frac{4 H_0 t^2}{T^2} $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "aafc2169",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle H_{0} - \\frac{4 H_{0} t^{2}}{T^{2}}$"
      ],
      "text/plain": [
       "           2\n",
       "     4⋅H₀⋅t \n",
       "H₀ - ───────\n",
       "         2  \n",
       "        T   "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xx = H0 - 4 * H0 * t**2/T**2; xx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "27f4cf37",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{8 H_{0} t}{T^{2}}$"
      ],
      "text/plain": [
       "-8⋅H₀⋅t \n",
       "────────\n",
       "    2   \n",
       "   T    "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vv = diff(xx, t); vv"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0cd95a21",
   "metadata": {},
   "source": [
    "$V_0$ (initial velocity) can be calculated by substituting -T/2 into the above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "fe4662ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{4 H_{0}}{T}$"
      ],
      "text/plain": [
       "4⋅H₀\n",
       "────\n",
       " T  "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vv.subs(t, -T/2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "61dc9122",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{32 H_{0}^{2} m t^{2}}{T^{4}} - g m \\left(H_{0} - \\frac{4 H_{0} t^{2}}{T^{2}}\\right)$"
      ],
      "text/plain": [
       "     2    2       ⎛           2⎞\n",
       "32⋅H₀ ⋅m⋅t        ⎜     4⋅H₀⋅t ⎟\n",
       "─────────── - g⋅m⋅⎜H₀ - ───────⎟\n",
       "      4           ⎜         2  ⎟\n",
       "     T            ⎝        T   ⎠"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ll = m*vv**2/2 - m*g*xx; ll"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9b9e7e69",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - H_{0} T g m + \\frac{32 H_{0}^{2} m + 4 H_{0} T^{2} g m}{12 T}$"
      ],
      "text/plain": [
       "                 2           2    \n",
       "            32⋅H₀ ⋅m + 4⋅H₀⋅T ⋅g⋅m\n",
       "-H₀⋅T⋅g⋅m + ──────────────────────\n",
       "                     12⋅T         "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ii = integrate(ll, (t, -T/2, T/2)); ii"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "6a0301fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{2 H_{0} m \\left(4 H_{0} - T^{2} g\\right)}{3 T}$"
      ],
      "text/plain": [
       "       ⎛        2  ⎞\n",
       "2⋅H₀⋅m⋅⎝4⋅H₀ - T ⋅g⎠\n",
       "────────────────────\n",
       "        3⋅T         "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(ii)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "fb069468",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle I = \\frac{2 H_{0} m \\left(4 H_{0} - T^{2} g\\right)}{3 T}$"
      ],
      "text/plain": [
       "           ⎛        2  ⎞\n",
       "    2⋅H₀⋅m⋅⎝4⋅H₀ - T ⋅g⎠\n",
       "I = ────────────────────\n",
       "            3⋅T         "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(Eq(I,simplify(ii)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "c86a47c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{\\frac{T^{2} g}{8}\\right\\}$"
      ],
      "text/plain": [
       "⎧ 2  ⎫\n",
       "⎪T ⋅g⎪\n",
       "⎨────⎬\n",
       "⎪ 8  ⎪\n",
       "⎩    ⎭"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(diff(ii, H0), H0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "491338c8",
   "metadata": {},
   "source": [
    "Took me a bit to figure this out but $ \\frac{T^2}{g} $ represents the value\n",
    "of H0 if we equated the kinetic energy as a function of H0 with the \n",
    "potential energy for H0 and solved for H0.  This is legitimate if we start with\n",
    "maximum kinetic energy and zero kinetic energy at the start and then at some point on\n",
    "the trajectory this is reversed."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af37449a",
   "metadata": {},
   "source": [
    "We are using a very crude graph with 3 points to see what happens when we apply the principle of least action.\n",
    "Specifically $ \\mathcal{L} = \\frac{1}{2}m v^2 - m g h $.  Because I am modelling with straight line segments I use $ \\mathcal{L} =  \\frac{1}{2}m v^2 - m g v t $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "bff9f5ab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{2 H_{0} \\left(\\frac{T}{2} + t\\right)}{T}, \\  H_{0} - \\frac{2 H_{0} t}{T}\\right)$"
      ],
      "text/plain": [
       "⎛     ⎛T    ⎞             ⎞\n",
       "⎜2⋅H₀⋅⎜─ + t⎟             ⎟\n",
       "⎜     ⎝2    ⎠       2⋅H₀⋅t⎟\n",
       "⎜────────────, H₀ - ──────⎟\n",
       "⎝     T               T   ⎠"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# First of all ascertain a piecewise function for x as a function of t\n",
    "# H0 is our highest point reached and T is the time taken for a particle for the entire trajectory, \n",
    "# leaving and returning\n",
    "x1 = (t+T/2)*H0/(T/2); x2 = H0 - (t)*H0/(T/2)\n",
    "x = (x1,x2)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "c895e2d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{2 H_{0}}{T}, \\  - \\frac{2 H_{0}}{T}\\right]$"
      ],
      "text/plain": [
       "⎡2⋅H₀  -2⋅H₀ ⎤\n",
       "⎢────, ──────⎥\n",
       "⎣ T      T   ⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now derive piecewise velocity function\n",
    "v = [diff(z,t) for z in x]\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "70bca4d8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{2 H_{0}^{2} m}{T^{2}} + \\frac{2 H_{0} g m t}{T}, \\  \\frac{2 H_{0}^{2} m}{T^{2}} - \\frac{2 H_{0} g m t}{T}\\right)$"
      ],
      "text/plain": [
       "⎛    2                     2               ⎞\n",
       "⎜2⋅H₀ ⋅m   2⋅H₀⋅g⋅m⋅t  2⋅H₀ ⋅m   2⋅H₀⋅g⋅m⋅t⎟\n",
       "⎜─────── + ──────────, ─────── - ──────────⎟\n",
       "⎜    2         T           2         T     ⎟\n",
       "⎝   T                     T                ⎠"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# compute the piecewise Lagrangian\n",
    "L1 = m*v[0]**2/2 + m*g*v[0]*t ; L2 = m*v[1]**2/2 + m*g*v[1]*t \n",
    "L1, L2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "bb11517a",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{H_{0} m \\left(4 H_{0} - T^{2} g\\right)}{2 T}$"
      ],
      "text/plain": [
       "     ⎛        2  ⎞\n",
       "H₀⋅m⋅⎝4⋅H₀ - T ⋅g⎠\n",
       "──────────────────\n",
       "       2⋅T        "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Integrate the Lagrangian over the piecewise interval w.r.t time (t)\n",
    "I1 = integrate(L1, (t, 0, T/2))\n",
    "I2 = integrate(L2, (t, T/2, T))\n",
    "II = simplify(I1+I2);\n",
    "II"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "8261bf1f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{8 H_{0} m - T^{2} g m}{2 T}$"
      ],
      "text/plain": [
       "          2    \n",
       "8⋅H₀⋅m - T ⋅g⋅m\n",
       "───────────────\n",
       "      2⋅T      "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cancel(diff(II, H0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "805d6de5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{\\frac{T^{2} g}{8}\\right\\}$"
      ],
      "text/plain": [
       "⎧ 2  ⎫\n",
       "⎪T ⋅g⎪\n",
       "⎨────⎬\n",
       "⎪ 8  ⎪\n",
       "⎩    ⎭"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now we want to find a stationary point w.r.t H0 (our highest point) and then solve for H0\n",
    "H0res =solveset(diff(II, H0), H0)\n",
    "H0res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "86d5d136",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\frac{T^{3} g^{2} m}{32}$"
      ],
      "text/plain": [
       "  3  2   \n",
       "-T ⋅g ⋅m \n",
       "─────────\n",
       "    32   "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "II.subs(H0, H0res.args[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a888cda",
   "metadata": {},
   "source": [
    "We are not going to plot the trajectory in matplotlib so we can see the basic shape of the trajectory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "42ff9334",
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = lambdify(t, x1.subs([(H0,1), (T,1)]), \"numpy\" )\n",
    "f2 = lambdify(t, x2.subs([(H0,1), (T,1)]), \"numpy\" )\n",
    "def f(t):\n",
    "    if t <= .5:\n",
    "        return f1(t)\n",
    "    else:\n",
    "        return f2(t)\n",
    "vf = np.vectorize(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "ba6a12a4",
   "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": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "xx = np.linspace(0, 1, 1000)\n",
    "\n",
    "ax.plot(xx, vf(xx))\n",
    "ax.set_title(\"Basic Proposed Trajectory Plot\")\n",
    "ax.set_xlabel(\"t (in 1/T units)\")\n",
    "ax.set_ylabel(\"H (in 1/H0 units)\")\n",
    "ax.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6523a83f",
   "metadata": {},
   "source": [
    "Now lets split the trajectory up into four line segments and we will call the height of the trajectory midway \n",
    "between half time and the start as H1.  (It's basically quarter time or T/4).  Again we break it up into a piecewise function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "42f50fd2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{4 H_{1} t}{T}, \\  H_{1} + \\frac{4 \\left(H_{0} - H_{1}\\right) \\left(- \\frac{T}{4} + t\\right)}{T}, \\  H_{0} - \\frac{4 \\left(H_{0} - H_{1}\\right) \\left(- \\frac{T}{2} + t\\right)}{T}, \\  H_{1} - \\frac{4 H_{1} \\left(- \\frac{3 T}{4} + t\\right)}{T}\\right)$"
      ],
      "text/plain": [
       "⎛                         ⎛  T    ⎞                   ⎛  T    ⎞            ⎛  \n",
       "⎜             4⋅(H₀ - H₁)⋅⎜- ─ + t⎟       4⋅(H₀ - H₁)⋅⎜- ─ + t⎟       4⋅H₁⋅⎜- \n",
       "⎜4⋅H₁⋅t                   ⎝  4    ⎠                   ⎝  2    ⎠            ⎝  \n",
       "⎜──────, H₁ + ─────────────────────, H₀ - ─────────────────────, H₁ - ────────\n",
       "⎝  T                    T                           T                        T\n",
       "\n",
       "3⋅T    ⎞⎞\n",
       "─── + t⎟⎟\n",
       " 4     ⎠⎟\n",
       "────────⎟\n",
       "        ⎠"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# position as a function of time as a piecewise function\n",
    "x1 = t*H1/(T/4); x2 = H1 + (t - T/4)*(H0-H1)/(T/4)\n",
    "x3 = H0 - (t-T/2)*(H0-H1)/(T/4)\n",
    "x4 = H1 - (t - 3*T/4)*H1/(T/4)\n",
    "x = (x1,x2,x3, x4)\n",
    "\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "68b72b97",
   "metadata": {},
   "source": [
    "Now we show what such a curve might look like by arbitrarily assigning values to H0 and H1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "b310b0bf",
   "metadata": {},
   "outputs": [],
   "source": [
    "ffs = []\n",
    "for xx in x:\n",
    "    fi = lambdify(t, xx.subs([(H0,1), (H1, .8), (T,1)]), \"numpy\" )\n",
    "    ffs.append(fi)\n",
    "\n",
    "def f(t):\n",
    "    if t <= .25:\n",
    "        return ffs[0](t)\n",
    "    elif .25 < t <= .5:\n",
    "        return ffs[1](t)\n",
    "    elif .5 < t <= .75:\n",
    "        return ffs[2](t)\n",
    "    else:\n",
    "        return ffs[3](t)\n",
    "vf = np.vectorize(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "a340cde7",
   "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": [
    "fig, ax = plt.subplots()\n",
    "\n",
    "xx = np.linspace(0, 1, 1000)\n",
    "\n",
    "ax.plot(xx, vf(xx))\n",
    "ax.set_title(\"Basic Proposed Trajectory Plot\")\n",
    "ax.set_xlabel(\"t\")\n",
    "ax.set_ylabel(\"H0\")\n",
    "ax.grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "f1eb8ae2",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{4 H_{1}}{T}, \\  \\frac{4 \\left(H_{0} - H_{1}\\right)}{T}, \\  - \\frac{4 \\left(H_{0} - H_{1}\\right)}{T}, \\  - \\frac{4 H_{1}}{T}\\right]$"
      ],
      "text/plain": [
       "⎡4⋅H₁  4⋅(H₀ - H₁)  -4⋅(H₀ - H₁)   -4⋅H₁ ⎤\n",
       "⎢────, ───────────, ─────────────, ──────⎥\n",
       "⎣ T         T             T          T   ⎦"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute velocities of our new piecewise function\n",
    "v = [diff(z,t) for z in x]\n",
    "v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "0dfae4f9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{8 H_{1}^{2} m}{T^{2}} + \\frac{4 H_{1} g m t}{T}, \\  \\frac{4 g m t \\left(H_{0} - H_{1}\\right)}{T} + \\frac{8 m \\left(H_{0} - H_{1}\\right)^{2}}{T^{2}}, \\  - \\frac{4 g m t \\left(H_{0} - H_{1}\\right)}{T} + \\frac{8 m \\left(H_{0} - H_{1}\\right)^{2}}{T^{2}}, \\  \\frac{8 H_{1}^{2} m}{T^{2}} - \\frac{4 H_{1} g m t}{T}\\right]$"
      ],
      "text/plain": [
       "⎡    2                                                  2                     \n",
       "⎢8⋅H₁ ⋅m   4⋅H₁⋅g⋅m⋅t  4⋅g⋅m⋅t⋅(H₀ - H₁)   8⋅m⋅(H₀ - H₁)     4⋅g⋅m⋅t⋅(H₀ - H₁)\n",
       "⎢─────── + ──────────, ───────────────── + ──────────────, - ─────────────────\n",
       "⎢    2         T               T                  2                  T        \n",
       "⎣   T                                            T                            \n",
       "\n",
       "                2      2               ⎤\n",
       "   8⋅m⋅(H₀ - H₁)   8⋅H₁ ⋅m   4⋅H₁⋅g⋅m⋅t⎥\n",
       " + ──────────────, ─────── - ──────────⎥\n",
       "          2            2         T     ⎥\n",
       "         T            T                ⎦"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the Lagrangians of our new piecewise function\n",
    "LL = []\n",
    "for i,xx in enumerate(x):\n",
    "    ll = m*v[i]**2/2 + m*g*v[i]*t\n",
    "    LL.append(ll)\n",
    "LL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "5a3fde31",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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wG/DgqguBi4oEfYRllcyvGzWKEvi/W1HouTSzB6h9dOIxeIRq1taQIUdH1ygTyL8juaJL9yBmImwPWYRvC0k0xA8Dr9sCmL9CDyPS8u/E99N9j8r6rtCFCPx6AXBKtwZJdXqNEoCwxWLQekI3w8yuxaPGzsI3x05iy9AQK/Q+JuJbp/rwk6oU/m55Pa5ChQ7gMOA/u1VIQkkWZYUKFSpUqNCr+H/xsPvsmCfQnwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(16 H_{0}^{2} - 32 H_{0} H_{1} + 32 H_{1}^{2} - T^{2} g \\left(H_{0} + 2 H_{1}\\right)\\right)}{4 T}$"
      ],
      "text/plain": [
       "  ⎛     2                   2    2              ⎞\n",
       "m⋅⎝16⋅H₀  - 32⋅H₀⋅H₁ + 32⋅H₁  - T ⋅g⋅(H₀ + 2⋅H₁)⎠\n",
       "─────────────────────────────────────────────────\n",
       "                       4⋅T                       "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IIS = []\n",
    "for i, xx in enumerate(x):\n",
    "    II = integrate(LL[i], (t, i*T/4, (i+1)*T/4))\n",
    "    IIS.append(II)\n",
    "action = simplify(sum(IIS))\n",
    "action"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "005edeaa",
   "metadata": {},
   "source": [
    "Now we substitute H1 for a scale multiple K of H0 and try to figure out K"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "d2a0048e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(32 H_{0}^{2} k^{2} - 32 H_{0}^{2} k + 16 H_{0}^{2} - T^{2} g \\left(2 H_{0} k + H_{0}\\right)\\right)}{4 T}$"
      ],
      "text/plain": [
       "  ⎛     2  2        2          2    2                ⎞\n",
       "m⋅⎝32⋅H₀ ⋅k  - 32⋅H₀ ⋅k + 16⋅H₀  - T ⋅g⋅(2⋅H₀⋅k + H₀)⎠\n",
       "──────────────────────────────────────────────────────\n",
       "                         4⋅T                          "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "aa = action.subs(H1, k*H0)\n",
    "aa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "6bca5f8a",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATIAAAAaCAYAAAA0TtAqAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAABJ0AAASdAHeZh94AAAMTUlEQVR4nO2dd7AV9RXHP8hTSDAYLGAZ66goQhQwsRFBETRqVIyOcZTEjB0bFpLYOHxVIjoRZWIHRxE1Y49hsBB0DBo7PoIJAVuAJI4oYMGCKJA/zm9h2bd7397LvfseeL8zd368/dVzdn/nd9oubVasWEEdddRRx9qM9Vp6AXXUUUcda4qGoiaS1NbMlhU1XzUg6WLgaKAr8BXwEnCxmf2jRRe2DqDO2+qjVjxdG/ZuIRqZpAZgSBFzVRn9gJuBfYEDgW+AKZI2bslFrSPoR5231UY/asPTn0jacQ3HqCmK0siuBf5Q0FxVg5kdHP9b0mDgE2A/YGKi7nxgNHCCmd1X2CLXUpTD2zryoYY8nQSMk3SBmX2yBuPUDDUXZJL6A5+Z2b9L1J8N7AN0AhYCbwBjzOzxZsY+EZgQ/jzVzMZltJsMDAAOMLNnM9rcCpwOnG5mt2dM+T1ci/0opa53KF8rteaiIOkaYE9gZ2BT4EtgLvAn4EYzWxhruwkwCDgM6AFsBSzF78OdwJ1mtjxjniJ4WzO0EtrT+n2I37e8GGJmtySuVYWnZrZC0i2AgKHNtZfUBXgPuNnMzkmpL7lvK6G9CNPyKuCOtApJ1wJT8A33Z+A6XPpvhqvJmZC0NXAj8FmONfQGlgGvlmizVyhLCaIxwHTgxZS6XsCnwFs51lMEzgc6AH/B130vbmqMAGYE/kU4FhiL8+Bl4AbgYaA7MA54QFKbjHmK4G0t0RpoXw2SOgA34YIj+o0M1UsT16PfEylDVY2nZvYasK+kbXI0PxIXoI8kK5rbt5XSXlONTNJhwBIzm5tSdyowDBgPnGZmSxP165cYtw1+Wi7EmXVRibY7ABsDM8zs84w238Uf3K/wkzitzWigD9An6fgMzO8KPGdmrSWfpaOZLUlelDQSuAS4mFV+yzeBI4BJce1D0iXAK8DPcCfyw4mxas7bAtCitKchjDciMdbuwKXAG2Y2IqVbcu5a8PR+/Nk5o5l2g/C9OTWxpmb3baW0NyQ6HAA8g2tG9wHDgf2BtuH6mWb2vqRuwGXAQUA74DlctZ2XGP+kJDFhnna4lJ1HihALBH2dtuCAc3FnZr9QlsKeoXy5RJveOC8a0+aVdD3wc9yEeDel/x74CTQt0a8TcBe+UcYAw5qhq2pIE2IBD+AP406xts9kjPF+MI1G4rx+ONGkCN7WFC1Je5mIXBfTSrYiP08ltcdNxV8AOwDzgduAa4CPgTlm1iPWZSpwuaTzzOyrjDE74nvyvhQBWs6+jaNZ2pMaWa9Q7gw8j6urdwCHAEcB7YKt/EdcsI0PizoMuJuYOSipbeh3b8q8A3Dz8QZgedDcugNLgFfMLFMVlrQrMAr3oU2VVA1Btncom6j/ksYAx+EPxayM/hHfVjJa0l74CdYJONbMHmpmnUXhp6GckbN9tAG/SakrgrctiZrRXgGi+UoKsrw8DVbElLDGafhB2wVXXnYFNgReT3R7I1zvAzydMfThwAbAo4n5yt23cTRLe5Yg+xGwt5nNCIu4AteeBgI9gQGRsJG0AfA2sL+k9jFNoAdOdNqG+WEolwCNuBBbCUlTgWPM7MPE9QbcSTgP1yryIGLCcUHjTEOqH0PSTcBgXIh/JGnzUPWZmcVt/NVODEkX4DdtFs6rFvObSboIvw8b4bzog9+TUTn6NuCnNcCTKU2K4G2LoJa0V4hmg0ll8vQ2XIgNB66KXCKSJuACDhKCzMyWSHoTD8xlCbJBwOfA5Ni6Ktm3cTRLe5YgOykSYgBmtljSHNyEGhbXmMxsqaTZwNa4czkSZF1DuZowCugcymHATODHuFNye+D3uMB8kKYO/+G4IO1jZl9mERUh2OQRTQOaa09TRkU+pORNE6vb8b2AxcACSY/hpuR43BRvdp01xkX4SRvhSfz+pt2XJEbhh8zjZvZUvKJA3rYUakl7WQiC4Ae4s7tUcmsunkraGzgBeMzMrow3NLOnJc0DtqGpRgawALfY0tbZHrfCnki4Nsrat4kxc9HeEOvQISzwXTNLO4G2BRbh5lJa3eJ4SB/YElhOenQiipZ+AxxhZnPC329IGgTMBvpK2iem+e2FS/PrSpmeCeyEayIvmdk+aQ2CH2sRnp7wz3idmWVFq+L92+Oq+Af4je9MiVSQjDHm4DzMi3vN7MQ8Dc1s8zBHFzxRchTQKOlwM0t7UKM1nQtciGuVg1Oa1Jy3JdY2hxrxK4xfU9orwG5Ae2Bamj85Qhk8PSuUIzPqF+KKyfSUuk/xvZ2GAbj2v9KsrHDfxpGL9rhGtjsuYKYkG0naDvf1PJJ0WkraENgR96nF0QFXadOieB+HsjEmxAAwsy8kPQWcjJu4LwapfDceYbo8i5gUROp/5oZl1ck6vcLozu44H9vhN/iecoRYwDus0mTz4L0yx8fM5gOPSnod5+PdJEz6CJLOxn0mM4H+ZrYopVkRvM1CzfhVBO2ShuDWyBa4kBtqZs+VGCu3oz8nBuLCKktT3AJ4y8wWp9R9AmyS0e9oXHOaBCu1qUr2bRy5aI8LsiYO65yD9QTa0PSmLsejnWmYHcqPM+qjBL7vhHJDVqmzSySl9RkraSzuTBwarpXzwFWq/kf9LwSOAU6UNM3Mbsg7gJn1r3DusmFmcyXNBPaQtKmZLYjXSxoKXI+r8f3N7IOMoYrgbSpqxa8iaJd0HC4oh+CH/xDgCUndUqL+yfnWWJAFC6IzLmCbKBmSdgE2B57NGKIBz51L9muLB5KesVXZ/5Xu2zhy0Z4myNIeulLOtp6hTN7UxUAHSeulZEY/DawAumXUR5pC9DbAV2Qk1YZ198QfitmsnvyX54GL1p8WVctzcsZ58xDwV+A6SfPMrElCYCtBZBokNYXf4KbndDxIsYBsFMHbwlAg7RcAd5nZ2PD3OZIOAc7Ec/vSUM23RpaFX6eM+l+HMou2jvjeTmJ/XFOLRysr3bdx5KI9KciyHGqlNLJIACYJj06XjiQ0r6AVTMSd4ufhpyAAkgYCB4c+T4b2XwKnpBEgaQTOkPFxk07SeuH6Ukr7KNb05OyF+0FmmdkySYfjXx24R9KBZvZSiblrAkk7A/Mt8V5c4MmV+In8gpl9FKu7HLgCv8cDM0yq+DhF8LYQFEV7iPD3xgNacUzG/Zdp8+V19OeCmX0dgnPdJPU3s5WBAUlnAb8Kf5YSZGnrGIRbYY/F5ip73yba5Ka9IXRoB3TDs5TTHGq9gbkJZ36EaCP/K3F9Zii3It2EPCsQMjrkkTXiUcuj8BPjlORGLBO74KptY5aTMPj3dsIDEsmcm2ZPzvBgdg9zLIOViZSHAn8DJoaAxdtrQEclOBS4WtLzuFa7EI9c9sUTH98HTo0aS/olvpGX4cnN56aYAXPM7K7w75rztigUTPumuLtlfqLLfDy5PA25nN1l4mo8HWKSpPvx56FPWPMsPOMgS5BtiSdVr0SI4h4FvBh8sdVCbtqj6GF3YH1SNC5J2+IqY1pdOzxiNyPp0Ay5Ux/gKRtNYGb/xQXkjTgDz8PTLSYC+5lZMpO6XORR/6MAR6Ot/npKdHJOTrRPnpw9cL41xhuZ2Uzc8bkRrmmU8wJsNTAFV+k3C+sYhr9qswgPxe8W1hhh+1C2xTO9LeV3Uqx9EbwtCoXRXiGq7ejHzO7Bs+z/BxwfftPx/M7If9bkRXNJ3we2o2lgb088ylltV0pu2tvU8lPXkm4HPjWzzHchWyMkbYnf5L5mNjV2fTj+mZ6umZ3rKIlvM2+DEP8CON7MHoxdvwnobmZ9W2xxvo7BeJRxmJklzV8k9cPzI7eLBwok/Q7XpHewjK/c1Bq1/vrFbbi/q446vvUI5tE0mibRDgBeKGINktpK6pxy/SDgFuA/wK0Z3Q8BxqVEOwcBf28pIQY1/h6ZmU2TNFvSnuafAVlbsAD3mXRJXO+C+xPqqBzfdt6OBiZIegX3o56B+52yhEe10Q14NeRqvoO7RvbAfWQfAkdayitiIchxMB6dXA1mtmstF5wHRXyP7FLgtALmqRpaw8m5ruLbzlszux/3xV2G+6X6AIdayqeuaoSlwOO4X2sInni+Mf4V5x5m1pjR70g8wpiWetHiqKmPLIL822OzWjJPqFyEFIEJ+M2OTs6TcUd5UQ/dOok6b9cuhCjsNcA5VQhe1ASF/OcjIczexC5vzWgFJ+c6izpv1zrsC/y2tQoxKEgjq6OOOuqoJf4PgMARbgW7FdwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(64 H_{0}^{2} k - 32 H_{0}^{2} - 2 H_{0} T^{2} g\\right)}{4 T}$"
      ],
      "text/plain": [
       "  ⎛     2          2         2  ⎞\n",
       "m⋅⎝64⋅H₀ ⋅k - 32⋅H₀  - 2⋅H₀⋅T ⋅g⎠\n",
       "─────────────────────────────────\n",
       "               4⋅T               "
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now try to find out where the action is stationary while we vary K\n",
    "diff(aa,k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "ec53225a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{16 H_{0}^{2} + H_{0} T^{2} g}{32 H_{0}^{2}}$"
      ],
      "text/plain": [
       "     2       2  \n",
       "16⋅H₀  + H₀⋅T ⋅g\n",
       "────────────────\n",
       "          2     \n",
       "     32⋅H₀      "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solve for k at the stationary point\n",
    "k_expr = solveset(diff(aa, k),k).args[0]; k_expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "b846c096",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVQAAAAaCAYAAADot6QdAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAABJ0AAASdAHeZh94AAANi0lEQVR4nO2dd5QX1RXHP8jqkizBWCKKR8EWBEFqVJQABhFbVCyHGBtGsaACUTGWyPUa+4koiSgoOUgQc0yMYgy2iBg0wYYQUKRExRIjCgiCikQgf9w3ML6dmd/Mb3+F3f19z/md2Xkzr9xX7rvtzTbZuHEjFVRQQQUV1B1blbsBFVRQQQUNBVWlqkhVm4rI+lLVVy6o6pXACUBb4CvgReBKEXm9rA1rRGjMY+BovxEYIyIXFbDMxP5sLOs7F0oioapqFTCkFHVtAegD3AUcDPwI+Bp4RlW3L2ejGhn60AjHQFUPAs4F5ha46D7k7s8jVXXvAtdb71AqCfVW4LclqqusEJH+4XtVPR1YBRwCPFaWRjUyNMYxUNVtgcnAzwApZNkp+3MqMF5VLxGRVYWsvz6h6AxVVfsCa0TknVDaDsAA4GigI7ArsA6YB0wAJojIhoiylgCtY6paKiI752jHRUAPYDtguatvtIg87r37NNAPOFREnospbyxwHnCeiNwTVy/wHUwT+DThnaIiC+1evtOASe52sIiMj3mvkP3l5/sE2DHt+8AQEbnbSyv7GKRFHei9B3hIRKarakEZagRq9aeIbFTVuwEFhidlVtWWwIfAXSJysUs7CegNdAY6uTomi8hpMWXky0OOBoYB7YEdgP8Cs4BRIjLTezfzWJRCQr0e+ImXdjJwN0bMdOA9oCVmpxmPqQ8ni0hUCMIq4I6I9DVxDVDVW4ERwAfAX4BlwPeAbpg64zOVbsB64JV4sjjQXV9NeAdgNDAHmJnjvaIgD9qDfLsBd2L92jxHNYXsr3AbaoAxXnIVcDW2eG6KyPZERFpZxyAt8qVXVQcDewORzKcIiOxPEXlVVe9S1d1F5L2E/MdhDPnhUNovMUa6Bpur++ZoQ2Yeoqq3AJdjAsUUbC3s7dpzoqqeISL3u3fzGouiMlS3G6wVkXe9R4uAY4Gp4V1EVa8CXgZOxDrmzxHFrhSRazO0YTDGUCYC54rIOu/51t79nsD2wFwR+TymzG8DHTAD/byEukcBPYGe5TDYZ6U9lN4E2+WXY5P+soQ6CtZfPlx513pldcIm9bw086DcY5AF+dCrqm0xJ1RPEflfsduYoj8fBK4Czk8oZgA2t2aE0n6OMdJ/Y5Lq9BxNycRDVHVnbB4vBfYXkY9DeQ4FngWuA+6H/OdelZchKPg24AFgJNALaOrSLxCRj1S1PbajHAZUA89jqoe/Kw3im52Ga+yzUY1xZY8FbsCkpyiGmhqqWu3Keo8IhuLq9Cdhd3d9KaHobljfzY6bxKp6OyaZHyoib2dte12RJ+0BhmLOhz7umoSC9FcGdHPXWbleTDsGqtoMU1PPAPbEFt044BZgJbBERDrWqdX5Ixe9PTC19A1VDdKaAr1U9XygRkS+CmfIl96U/TkDuEZVh/n1ujJaYHPqgTBDFpHpoXdiit6MPHhIa0wqfinMTIO6VXU1prklIefc8yXUru76feAFTH36HXAEcDxQ7ewkf8AY7ETX6KOB37u/AQujcPkm52ikj2DBfR3zvNrZ9nYHPsc8mjNidst+WCfdAWxwEnMHYC3wsm8zcUjDIA5y10j1VVVHAwOxibcgoZxiIh/aUdV2wM2YfXWGqhaCoSb2V0YE9SUy1LRj4FS7Z1wbZ2HqbEtMmGiHmTteq3uz80YueqdQu18nAIsxydXXSvKiN8OcnufK6AlMi3h+DLAN8EhCGXVFFA9ZjPXFAaq6o4gsCx6oai/MZjslR7k5514cQz0AOEhE5roKr8MkncOBLkC/YEGq6jaYmN5LVZuJyFpXRkesY1OHcLjwqjPc7ZMxr+3MZkdJgHdU9SwR+buX/gN3XQvMxhhKuL4ZwEki8kkoOei0gU5ij0KsPVBVxwCnYxvQp07VAHPMxdp5i4DMtLv+n4SN9VUp66lTf+WBQEqILSvjGIzDmMtI4PrA5qaqkzDGA+VlqIn0ishKTKrcBFX9HFgRE3ebmd4s/Skia1V1ESY5RzHUAZgg9HQUPXVFHA8RkRWq+gtgFDBfVadgZoe9MNPB3zCnaRJyzj0/DjVgqIMCZuoasxpYgqkSI8LSjVMlFwJNgJpQWW3dNcyscuFmbOE/LiJPRTyfAPTFmGoNxrTHAW0w43wn7/2d3HUEsBH4IbYT7Y8NaC/gT8HLznYY9EE/4NSYXxBvF9WxQ1wd0zCDefCLtUMWCZlodxiJbZiDROTLXBUUqL9Swy2W/TFJIylIP9UYuLjNU4FHReRXYQeGiEzDNhYoE0PNQG/a8vKlN+ucXoZpuX79zTCt9YmQ4FVoxPIQEbkDs6tWAYOBKzDn1vvAfb4pIIy0Y1EVylCDdcLbIhIlHbYGVmBG56hnq0VkeSitFbCBBO+71+ChwKXAAmw3rAUR8Y0rrwPnq+oal/dabAcMEGwYXwPHisgSdz9PVQdgG0FvVe3hNol9gG2BF0WkR0w7t8P64UvgjYg2NkmmNBo5QsKiEBtS4pCJdlU9EJNKb4szB0Sgzv2VEfsBzYBZUTbhABnG4EJ3vSHm+XJgN8yjHYkijFsYqej1ISJ9Yh7lRW8ec/ozbP376IdprUVR93PxEFW9HDOD/AaLYPkIiya4CZisqp1F5PKY4lONRVjl74Qtwmf8l1S1DRa/+LDvVFDV5pgE8oKXrQZTCXJ+fUVVL8JsOfOBviKyIlceD2Oxjuzlpa9019khhgKAiHyhqk8BZ2MmjplsVl+TJJJAIptTYK/xW5h6nhYf5ni+0l1z0q6qr2A28EXANRnaUKf+UtUhmAS9C8Zsh4vI8wllpXZIpcThGBOJk5x3ARY7DS0OhR63MLZEetNgFRbj6eMETMKbWsfyayEXD1HVPpjT7RERuST06DUnYCwCLlXVsTEOt1RjEWaowcSPypBUWBdM3fcX1QbMRJAIVR0O3I5Jm32TxO4EBGaFGi99obuujMkXBCZ/y12zMIhC2AM3QUT6FrI8stHenM0q2toYL+u9qnov5qwa7tLy7i9VHYgtgCHYZjwEM9u0T4hhTOWQSgOnfu6EMfpam76q7ouZlp5LKqcI4xbGFkdvSlRhccnh8psCPwaelQKfpErJQ45x11rhWE7AeBnTbrsAUQw11VhEMdQoRpFkjO3irv6iWg3UqOpWEnFiAcAZiW/GVIx+Yc9bRgReZL8jpmH2w/Yx7QgcNcEprjQMIqA3yiGVVeIqJrLQ/hUWzRGFrhjNL2BMOmwOqEt/XYLZre519xer6hHABcCVMWXldApkwHr32y7meaD6bbEOqYwoJb0tsPUfRi9Mai2oup+Bh1S7a1xoVJAep86nGouwU6or8QbXJAk1YMT+QARSRouoilX1GqwjZmG7SiIzVdV2zs7rp7fB7CHggnIDiB0oeAwLsRrm5Tsc6I9JcE+q6lbY4l9Hsq0vl8R1oyvnn5jEtXsSXcVCFtpF5EsROSfqh52uApjo0h50ZeTdXy4ypBu1Pb1PYx/gqIVCO2ic6Woh0FrtaG64rguBs9xtg3BIlZjeFmxe/wEGYFrrowUoH8jMQwLB5lxV3dUr50jsuwRrsXXr15N6LKpchmrsbOvcGINrN+Bdz+kUoCvmcHjTS5/vrrtSO6zjTOxUwnqM0KERauYSEbkvdD8Qs3HMAN7FdsC9sBjYZtgRyl9HtO9CbOGPcrGYs4E9sBCQ9cA5IrJK7bBCc8zmGLlLOXvxPpijzY/Fy0fiKjZS0Z5n2fuSf3/tiJmDlnpZlmKHRaKQl4MmB27CwsSmquqDmJOip2vzAixSpVwSan2mtxXwx+DGRYMcD8wUEX/Mg3eOd++AmR4Aeqjqfe7vZSJyWej9rDzkIcw/dBjwpqo+gtHfDjMHNAGuiOFxqccikFA7AFsTIYGqamtMVI96Vu0aNNd3OIjIYuBj7GMHPvZw16bYiQ2J+A3y8kwH/oox0Z9iDKw3poqeCRwTRayIfIBtCHdiE2cYdgDhMeAQEQlOUqRRXwPH3Wz55nG3zBJXKZCB9nyQd3/liUI7aBA7tz0U+A9wivvNwWJ4A3tjuT6oUi/pVdXvYmGMYSd1dyx64OGILAE6Y+v4TEx7AjvFFaSd5L2fiYe4+XcUdsR1PiYxX4qZCx8H+ovI6Ji2pR6LJsX8Fyiqeg/wWXhnaYhQ1VbYJO0tIjNC6SOBU0WkbWzmRgi3AX0BnCIi4TjgMUAHEeldtsZZO07Hoh5GiEiU1tOgUEh6nTd9ItBGNh8auBHT0vaU0FfnGiKK/YHpcWzebSqoANh0GGQWFpcYRj8ibFjFgKo2VdWdItIPw75i9D4WjtcgUEJ6jwDGe5EEA4B/NXRmCkX+HqqIzFLVharaXUQKGma0hWEZZstp6aW3xOw0FdTGKGCSC1f5B/Z1olaUjom1B15x8bhvYSavzphN8RPgOCntUeFio+j0Okdlf7x4cBFpV5dy6xNK8S9Qrsb+LUODxZYgcdU3uGiB4dhXy+ZgC/soqf2px2JhHWY7647FwJ6NfYbwVqCjiMwuUTtKhVLQexwWDVLXgwH1FkW1oQZQ+y7ngjLGZBYdLmxqEjZZA4nrbGC/EjKJCiooC1xExy3AxQVwQNZblOSf9LlQolr2m4aELUDiqqCCcuJgLOyo0TJTKJGEWkEFFVTQGPB/NE6irj0fmX8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(256 H_{0}^{2} - 64 H_{0} T^{2} g - T^{4} g^{2}\\right)}{128 T}$"
      ],
      "text/plain": [
       "  ⎛      2          2      4  2⎞\n",
       "m⋅⎝256⋅H₀  - 64⋅H₀⋅T ⋅g - T ⋅g ⎠\n",
       "────────────────────────────────\n",
       "             128⋅T              "
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# subsitute k back into our original expression for action\n",
    "action = simplify(aa.subs(k, k_expr)); action"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "ba569eba",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(512 H_{0} - 64 T^{2} g\\right)}{128 T}$"
      ],
      "text/plain": [
       "  ⎛             2  ⎞\n",
       "m⋅⎝512⋅H₀ - 64⋅T ⋅g⎠\n",
       "────────────────────\n",
       "       128⋅T        "
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(action, H0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "2906ba7c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{\\frac{T^{2} g}{8}\\right\\}$"
      ],
      "text/plain": [
       "⎧ 2  ⎫\n",
       "⎪T ⋅g⎪\n",
       "⎨────⎬\n",
       "⎪ 8  ⎪\n",
       "⎩    ⎭"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(diff(action,H0), H0)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "821ec047",
   "metadata": {},
   "source": [
    "Now we get the same expression for H0 as before so now lets use this to calculate H1.  Remember we have an expression for k above"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "9736e451",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{3}{4}$"
      ],
      "text/plain": [
       "3/4"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "k_expr.subs(H0, H0res.args[0])\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "51188ce6",
   "metadata": {},
   "source": [
    "Hence we have found that H1 is 3/4 of H0.  This makes sense for an upside down parabola which reaches a height of 1 unit in the middle.  Therefore 1/4 of the way along it should reach a height of 3/4.  I am wondering if we can work out points on the true trajectory for all applications of the Lagrangian this way or just the one we used in this special case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "efd78021",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 0, \\  T_{\\alpha}, \\  \\frac{T}{2}, \\  T - T_{\\alpha}, \\  T\\right]$"
      ],
      "text/plain": [
       "⎡           T               ⎤\n",
       "⎢0, Tₐₗₚₕₐ, ─, T - Tₐₗₚₕₐ, T⎥\n",
       "⎣           2               ⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tintervals = [0, Ta, T/2, T-Ta, T]; tintervals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "8da7c756",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 0, \\  H_{\\alpha}, \\  H, \\  H_{\\alpha}, \\  0\\right]$"
      ],
      "text/plain": [
       "[0, Hₐₗₚₕₐ, H, Hₐₗₚₕₐ, 0]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yexprs = [0, Ha, H, Ha, 0]; yexprs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "7b7ab64a",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{H_{\\alpha}}{T_{\\alpha}}, \\  \\frac{H - H_{\\alpha}}{\\frac{T}{2} - T_{\\alpha}}, \\  \\frac{- H + H_{\\alpha}}{\\frac{T}{2} - T_{\\alpha}}, \\  - \\frac{H_{\\alpha}}{T_{\\alpha}}\\right]$"
      ],
      "text/plain": [
       "⎡Hₐₗₚₕₐ  H - Hₐₗₚₕₐ  -H + Hₐₗₚₕₐ  -Hₐₗₚₕₐ ⎤\n",
       "⎢──────, ──────────, ───────────, ────────⎥\n",
       "⎢Tₐₗₚₕₐ  T            T            Tₐₗₚₕₐ ⎥\n",
       "⎢        ─ - Tₐₗₚₕₐ   ─ - Tₐₗₚₕₐ          ⎥\n",
       "⎣        2            2                   ⎦"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vs = []\n",
    "for ii in range(len(tintervals)-1):\n",
    "    vs.append ((yexprs[ii+1] - yexprs[ii])/(tintervals[ii+1] - tintervals[ii]))\n",
    "vs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "4e506933",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{H_{\\alpha} t}{T_{\\alpha}}, \\  H_{\\alpha} + \\frac{\\left(H - H_{\\alpha}\\right) \\left(- T_{\\alpha} + t\\right)}{\\frac{T}{2} - T_{\\alpha}}, \\  H + \\frac{\\left(- H + H_{\\alpha}\\right) \\left(- \\frac{T}{2} + t\\right)}{\\frac{T}{2} - T_{\\alpha}}, \\  H_{\\alpha} - \\frac{H_{\\alpha} \\left(- T + T_{\\alpha} + t\\right)}{T_{\\alpha}}\\right]$"
      ],
      "text/plain": [
       "⎡                                                                 ⎛  T    ⎞   \n",
       "⎢                                                   (-H + Hₐₗₚₕₐ)⋅⎜- ─ + t⎟   \n",
       "⎢Hₐₗₚₕₐ⋅t           (H - Hₐₗₚₕₐ)⋅(-Tₐₗₚₕₐ + t)                    ⎝  2    ⎠   \n",
       "⎢────────, Hₐₗₚₕₐ + ──────────────────────────, H + ───────────────────────, H\n",
       "⎢ Tₐₗₚₕₐ                    T                              T                  \n",
       "⎢                           ─ - Tₐₗₚₕₐ                     ─ - Tₐₗₚₕₐ         \n",
       "⎣                           2                              2                  \n",
       "\n",
       "                                ⎤\n",
       "                                ⎥\n",
       "        Hₐₗₚₕₐ⋅(-T + Tₐₗₚₕₐ + t)⎥\n",
       "ₐₗₚₕₐ - ────────────────────────⎥\n",
       "                 Tₐₗₚₕₐ         ⎥\n",
       "                                ⎥\n",
       "                                ⎦"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs = []\n",
    "for ii in range(len(tintervals)-1):\n",
    "    xs.append (yexprs[ii] + vs[ii]*(t-tintervals[ii]))\n",
    "xs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "1591072a",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{H_{\\alpha}^{2} m}{2 T_{\\alpha}^{2}} + \\frac{H_{\\alpha} g m t}{T_{\\alpha}}, \\  \\frac{g m t \\left(H - H_{\\alpha}\\right)}{\\frac{T}{2} - T_{\\alpha}} + \\frac{m \\left(H - H_{\\alpha}\\right)^{2}}{2 \\left(\\frac{T}{2} - T_{\\alpha}\\right)^{2}}, \\  \\frac{g m t \\left(- H + H_{\\alpha}\\right)}{\\frac{T}{2} - T_{\\alpha}} + \\frac{m \\left(- H + H_{\\alpha}\\right)^{2}}{2 \\left(\\frac{T}{2} - T_{\\alpha}\\right)^{2}}, \\  \\frac{H_{\\alpha}^{2} m}{2 T_{\\alpha}^{2}} - \\frac{H_{\\alpha} g m t}{T_{\\alpha}}\\right]$"
      ],
      "text/plain": [
       "⎡      2                                                      2               \n",
       "⎢Hₐₗₚₕₐ ⋅m   Hₐₗₚₕₐ⋅g⋅m⋅t  g⋅m⋅t⋅(H - Hₐₗₚₕₐ)   m⋅(H - Hₐₗₚₕₐ)   g⋅m⋅t⋅(-H + H\n",
       "⎢───────── + ────────────, ────────────────── + ───────────────, ─────────────\n",
       "⎢        2      Tₐₗₚₕₐ         T                              2       T       \n",
       "⎢2⋅Tₐₗₚₕₐ                      ─ - Tₐₗₚₕₐ         ⎛T         ⎞        ─ - Tₐₗₚ\n",
       "⎢                              2                2⋅⎜─ - Tₐₗₚₕₐ⎟        2       \n",
       "⎣                                                 ⎝2         ⎠                \n",
       "\n",
       "                        2        2                 ⎤\n",
       "ₐₗₚₕₐ)   m⋅(-H + Hₐₗₚₕₐ)   Hₐₗₚₕₐ ⋅m   Hₐₗₚₕₐ⋅g⋅m⋅t⎥\n",
       "────── + ────────────────, ───────── - ────────────⎥\n",
       "                       2           2      Tₐₗₚₕₐ   ⎥\n",
       "ₕₐ         ⎛T         ⎞    2⋅Tₐₗₚₕₐ                ⎥\n",
       "         2⋅⎜─ - Tₐₗₚₕₐ⎟                            ⎥\n",
       "           ⎝2         ⎠                            ⎦"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the Lagrangians of our new piecewise function\n",
    "LL = []\n",
    "for i in range(0, len(v)):\n",
    "    ll = m*vs[i]**2/2 + m*g*vs[i]*t\n",
    "    LL.append(ll)\n",
    "    #display(ll)\n",
    "LL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "3b63b679",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAiIAAAAmCAYAAAD5uvAVAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAABJ0AAASdAHeZh94AAAN5UlEQVR4nO2de/AWVRnHP4ClKIyKlUhextHULOXmmCKSOKOM2h9emlBGsiwrSe0yVqJND08aacblp4NjmpNhjRROIpRpSkJoInkBxUujYhODyoiiiTIoSH+cfXFZdt89Z3ff6+/5zPzm977nnD37fJ9z2fOec3a3z9atWzEMwzAMw2gFO7XagDxUdT/gNuATwGbgShGZ21qrjGZidaB6eqtPVXUP4H5c37cT0CMiN7fUqJL0prLsTVrbhWb4vG+VmTWIzcB3ReRw4GRgpqru1mKbjOZidaB6eqtP3wbGiMgw4HPA5aq6V2tNKk1vKsvepLVdaLjP+1S1NKOqR4vIskoyq3+eFcAXRGR1o89ltCdWB6qnN/pUVQcBjwNHici6VttTFb2pLHuT1nYh1Oc+Y4NKlmZU9SzgmYy4ycBUYJaIXJSIWwisFZEJifAvAXOAPUXkrVj4SKBfzQGqejJwb45554nI7EBJ3qhqP2AKcC6wD/AK8HtgiohsjqXz1hrql0SalvskZkuejjOAeTnZbGdrvA60k9YQSrYJL58Bv81LU/NLsl21irJ9BW6GI7c+APOBxcCngB+0YhASaT0TOBTYBCwFJovIykS6jukjy/RbRWxtx+uBD40q+5Tz+Pgjj8z+NyCPXVT18yKyOCtB6aUZVR0AHCsiz6bEHQN8A3gy4/ARwKMp4UcBLyQa2CBgdpRfjSW4i3/tbzUwLRE2J1BSKD8Cvg1cAhwGfCf6PjmRzltrYNok7eCTGnV1AH8jwNaUOtBSrap6q6pOCTymVJvAz2d3eaSZE9mT1q6aTkV9hVd9EJE3RWQocCAwQVX3rkhDSH04AbgBGAWciJv+vj8qjzid1EeW6beCbG0DrTsQUP4nUHHZZ5DnjyHAAXXi8/pfL0TkH8A50Y/2VLbNiKjqvpEhZwPfBI4BngfGAwOA6TgHrAS+KCL/jQ69ENc5boeq7o6bGTgfkJT4g4A9yHb0Y7G0O+N+BV4tIv+MCdwIbIydb1/gIRF5NUtwAxgFLBCRBdH3/6jqfNyvMyLbQrR6p02jTXzipSPE1rQ6UJXWaJQ/HTgaeBHX0BYDJ4nIopC8cs5Tuk0EaH4rL01Wu4rFd4xfILw+iMjaaJr5eOCO2PkarltExsW/q+pEXJkdByyIwjqmjyzbb4G/rY3U2ollX+c8If1rcP8bi/Px2RPAWcAf02yNz4gMjf5fCCjuQrozcCvwC+By4FjgY8D3Y8edgZtaSnITcIeIPJB2YmAk8AFujTYuqg9uJPhY7PutwN9F5LaMvIiO6YNHAVXMg8BYVT0MQFUPx41y746l8dJaIG0erfIJhOvItNWzDhTSGpXbA8AjUR6XAX/ADdKXh+TlQSVtIoaP5tQ0eT7tcL9Atu69VXVg9Hl3YAzw71h8M3XHGYjrj9fHwjqpj6yy34KC9Tbv+Hp0YtkHkOePQv1vgM8exg1EUonvERmGG5WNF5G10Unuw82QHCYir0dhi3FTNkQNeh8R+V/CuAuAg3H7JrIYiXP+26qaFl9zyHG4WZknVfX0KGyiiDyVkt+62ExNs7gGV5GeUdUtOJ/+TERuSNjmozU0bR6t8knt3CE66tnqUweKar0OuE9Efhh9fzZagx0jIm8G5pVJxW0inj5Pc1aaPJ92sl9qx6TpPgC4Kepc+wDXJ+pRU3Sn0IPruB+OhXVSH1llv1XLr0i9zTu+Hp1Y9r7k+aNo/+vrs2dwS1KpxAciQ4E/1wYhEfsDf6oNQmJh/4o+DwHiJ0NVD8VtOBstIu9nnRg3evoL8ONE+GnAlUQjQRF5EL+9LCNw0z+FUNWrgCtyko1NmZ4bD3wZmAA8jRvQ9ajqSyJyS8y2XK0F0ubRKp/Uzh2iI9NWzzoQrFXd/fEn4cosziZgRUr6y3EzgzV2Braq6qWxsFNEZEniuErbRCJ9nubUNPV8GuqX6JjgutJAv9SOSdO9jB111expSn1IyWc6MBrnhy0JDU3tI0u0+Sr7rVp+QfXW5/gsCtb50uVftuwDyPNHcP8b4jMR2ayqH1XVj4rIe8m8kjMi1yXihwM/SYQNxU2lAuwFbEjE15Zvno6N5PoBY1T1W8BuIrIJJ/xqEVmeEDcBv404SUYAdyYDVXU4MAO3/jUVV5CTEgMugJnA73LOkTZavBb4pYjUNvU8paoH4Darxgcivlq90nrqSvVJwPEzKeYTbx0+tnoSrDU6ZjM7bpD8NLAwJasb2X6N8xpgDdu3mzUpxzWqTfj4rIhfQ/0CxepKI/uKZuguWh+2oaozcLPOY0VkVYo9De8jE8ykWJv3trVs3+VJ0PWAYnW+VPlXVfYV+bMZ7WUDbszwSjJiJwB1Dyc5iNiISN1DfvZLhO0XZVQLewvon8hzHjtusPkNbuPrVOA9VT0QGET22m7oOv8A4BB2XE/bDbgdOBV4F1gG3JUyCEHcLXxFbuPbFdiSCNtCNIIM0eqb1kdXlk98j4fiPgkt33q2ep6vkNZoKa0frgzfidKPxE1FTk/mJSJvAG/E8n4beENEXsgxcR4Vtwkfn5Xwa5BfoHBdmUcD+opm6S5RH2rpe3AzqmNF5LlEXFP6yCRFyjGwjyvVd3naE3w9CO0LoFz5V1X2Vfizif3ErkSb6JPUZkSOjP4vj8UNw02xPJ0I24C7jRDgVdzO3m1E60JvxsNU9R1cAa2MGQvpwocDP08ztg7Dceu9yfzGAUtEZFW0JjwYN4NRJQuAy1T1JZyvhuM2886O4kO0+qb10ZXlE9/jyxBavvVs9aGo1kdxdfxaVZ2Ge67EjChueUFbdqBBbcLHZ5lpcn5FdbJfauFtqzuyYxYwETgdWK+qg6OoDSKygYA25PmLuGwbq0dIGZXtu3wocj3oyLKnGn/Wja8zi+Tts8g2ROTdtHPU1n2GAs+LyDsJ41ZK7KFcUboVIvJBlOnrwEZV3SVDYBYjgVXJDUDRkkbWSLAeI3AjreT01pF8uFZ1BLC0ARu1Lsbd+ncD8CzuPuyb+XCdNUSrb1ofXVk+8T2+DKHlW89WHwppFXeb2nm4XxNPAhfhdoe/ljJV2mga4bPUNLFfUefj1qenAC/XLlwd7hfoDN2TcJvcF+Kmqmt/tf0FXrrzNMUo28bqEVJGZfsuH4KvB51Y9nmaYuT5MzO+Xv0K9NnBpN9dC1TwiHdVvRo3FfRwbuImE20mGoDb8HMn8HERGdVaq8pTVle3+iWNEK3RqP0e4EURmdQ8K5uLqp6J20R3QaR5E3Bw1mC0W/zSjbpDNbWaVvY93dgXNNqfIfWrns+ivS0DReRXaeep4hHv1+NmBdpuIILbdDUfd//yNGCiql4iIslNuZ1GWV3d6pc0MrWq6mjcVObjuL1P38MtP361RbY2i7q/orrYL92ouxmzvlXSyr6nG/uCRvszs34F+mwM2z9/bDtKD0REZI2qrlbVvdM2gbaSyGHDYkE3ZSTtKMrq6la/pJGjdTBup/sngdeARcBIEXm5Wfa1iPeAIaraF3crYLIf6Fa/dKPuPE1tRSv7nm7sC5rgz3r1y8tnqvoZYFHW/hCo6O270ZTMOBG5p3RmhmE0FFXdH/crqj/RryhgbpfOiG2jG3V3oyajfaiifqnqKSLy13ppKhmIGIZhGIZhFKFvfhLDMAzDMIzGYAMRwzAMwzBahg1EDMMwDMNoGbVHvNtGEcMwDMMwmoaI9AHbrGoYvRZV3RN4DhglIi+22p6qUNW5uOcdTGu1LYZh5NPW95wbRm9FVScDZwKH4p5muBSYLCIrVfVk4N6cLM4Tkdk5aS4H7haRFyvMM5d62qL4srb8FFisqr+W8DfUGobRZGyPiGG0Jyfg3l80CjgR97rt+1V1ELAE2Cf2txp3j388bE69zFV1V+DrwC1RUOk8K9JW2hYReQr33oxzK7LXMIwGYjMihtGGiMi4+HdVnYh7MdVxIrIA2BiF7457K+ZD0UuofDkV2Ao8FJ1vYwV5elFPG7CgIlvmA+cAsyox2jCMhmEzIobRGQzEtdf1ifARuFd4h76x+njgMRFJ2yRWNM+iZGkrY8sy4GhV7V/SNsMwGowNRAyjM+gBlrPjyyVHAusKvOjsACDrPRpF8yxKlrYytrwMfAQYUs40wzAajS3NGEabo6rTgdHAaBHZkogeATxRINv+QNZLKr3yVNWrgCtyko0VkUV18qinzduWFDZG/21GxDDaHBuIGEYbo6ozgLNxF/RVKUlGAHcWyHodsGdGnG+eM3GvIa9H5kyGh7YQW5LUNr6+VuBYwzCaiA1EDKNNUdUeYDzuQv1cSvwA4BDg8YzjhwMzcJs9pwKnAZNEZC1uluEroXnGEZF1uAFNMHna8mzJ0QbwWWBN7LthGG2KDUQMow1R1Vm4V26fDqxX1cFR1AYR2RB9Ho7byJl2od4NuB13d8y7uM2bd8UuzPcC16jqXiLyeuzQzDyrwlNbpi0e2sBtxs17FolhGG2AbVY1jPZkEu5ukoXAK7G/S2NpRuBue01b1hgHLImWPNYCg4Fra5HRszaW4ZZG4tTLsyp8tNWzpa42Vd0FOAO4uRHGG4ZRLTYjYhhtSO0dDDlpenB3nKRxJLAi+nwE7pHnyf0aCvSo6o21jaI5eVaCj7YcW/K0fQ14RESWljLUMIymYDMihtGdvAcMUdW+wJWk/OgQkXtwD/zat8m2lSVP2/vAxU23yjCMQthL7wyjC1HV/XFPF+2Pezz6RGCuiFzXUsMqoJu1GUZvxAYihmEYhmG0jP8Dw1dcmLQgBJIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(4 H^{2} T_{\\alpha} - 8 H H_{\\alpha} T_{\\alpha} - H T^{2} T_{\\alpha} g + 4 H T T_{\\alpha}^{2} g - 4 H T_{\\alpha}^{3} g + 2 H_{\\alpha}^{2} T - H_{\\alpha} T^{2} T_{\\alpha} g + 2 H_{\\alpha} T T_{\\alpha}^{2} g\\right)}{2 T_{\\alpha} \\left(T - 2 T_{\\alpha}\\right)}$"
      ],
      "text/plain": [
       "  ⎛   2                                 2                        2            \n",
       "m⋅⎝4⋅H ⋅Tₐₗₚₕₐ - 8⋅H⋅Hₐₗₚₕₐ⋅Tₐₗₚₕₐ - H⋅T ⋅Tₐₗₚₕₐ⋅g + 4⋅H⋅T⋅Tₐₗₚₕₐ ⋅g - 4⋅H⋅Tₐₗ\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                            2⋅Tₐₗₚₕₐ⋅(T - 2⋅Tₐ\n",
       "\n",
       "   3             2             2                             2  ⎞\n",
       "ₚₕₐ ⋅g + 2⋅Hₐₗₚₕₐ ⋅T - Hₐₗₚₕₐ⋅T ⋅Tₐₗₚₕₐ⋅g + 2⋅Hₐₗₚₕₐ⋅T⋅Tₐₗₚₕₐ ⋅g⎠\n",
       "─────────────────────────────────────────────────────────────────\n",
       "ₗₚₕₐ)                                                            "
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the integral of the Lagrangian\n",
    "II = []\n",
    "for i in range(0, len(v)):\n",
    "    iii = integrate(LL[i], (t, tintervals[i], tintervals[i+1]))\n",
    "    II.append(iii)\n",
    "    #display(iii)\n",
    "res = simplify(sum(II))\n",
    "res"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "449bd670",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(4 H^{2} T l - 8 H H_{\\alpha} T l - 4 H T^{3} g l^{3} + 4 H T^{3} g l^{2} - H T^{3} g l + 2 H_{\\alpha}^{2} T + 2 H_{\\alpha} T^{3} g l^{2} - H_{\\alpha} T^{3} g l\\right)}{2 T l \\left(- 2 T l + T\\right)}$"
      ],
      "text/plain": [
       "  ⎛   2                             3    3        3    2      3               \n",
       "m⋅⎝4⋅H ⋅T⋅l - 8⋅H⋅Hₐₗₚₕₐ⋅T⋅l - 4⋅H⋅T ⋅g⋅l  + 4⋅H⋅T ⋅g⋅l  - H⋅T ⋅g⋅l + 2⋅Hₐₗₚₕₐ\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                                  2⋅T⋅l⋅(-2⋅T⋅l + T)          \n",
       "\n",
       "2               3    2           3    ⎞\n",
       " ⋅T + 2⋅Hₐₗₚₕₐ⋅T ⋅g⋅l  - Hₐₗₚₕₐ⋅T ⋅g⋅l⎠\n",
       "───────────────────────────────────────\n",
       "                                       "
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res1aaa = res.subs(Ta, l*T); res1aaa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "id": "69cc8179",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{m \\left(2 H^{2} T a^{2} - 8 H^{2} T a l + 4 H^{2} T l + 2 H T^{3} a g l^{2} - H T^{3} a g l - 4 H T^{3} g l^{3} + 4 H T^{3} g l^{2} - H T^{3} g l\\right)}{2 T l \\left(- 2 T l + T\\right)}$"
      ],
      "text/plain": [
       "  ⎛   2    2      2            2            3      2      3              3    \n",
       "m⋅⎝2⋅H ⋅T⋅a  - 8⋅H ⋅T⋅a⋅l + 4⋅H ⋅T⋅l + 2⋅H⋅T ⋅a⋅g⋅l  - H⋅T ⋅a⋅g⋅l - 4⋅H⋅T ⋅g⋅l\n",
       "──────────────────────────────────────────────────────────────────────────────\n",
       "                                            2⋅T⋅l⋅(-2⋅T⋅l + T)                \n",
       "\n",
       "3        3    2      3    ⎞\n",
       "  + 4⋅H⋅T ⋅g⋅l  - H⋅T ⋅g⋅l⎠\n",
       "───────────────────────────\n",
       "                           "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res1aa = res1aaa.subs(Ha, a*H); res1aa"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "id": "dffd62b3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - m \\left(4 H a^{2} - 16 H a l + 8 H l + 2 T^{2} a g l^{2} - T^{2} a g l - 4 T^{2} g l^{3} + 4 T^{2} g l^{2} - T^{2} g l\\right)$"
      ],
      "text/plain": [
       "   ⎛     2                         2      2    2            2    3      2    2\n",
       "-m⋅⎝4⋅H⋅a  - 16⋅H⋅a⋅l + 8⋅H⋅l + 2⋅T ⋅a⋅g⋅l  - T ⋅a⋅g⋅l - 4⋅T ⋅g⋅l  + 4⋅T ⋅g⋅l \n",
       "\n",
       "    2    ⎞\n",
       " - T ⋅g⋅l⎠"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res1a = cancel(diff(res1aa, H)).args[1]; factor(res1a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "id": "23fe3793",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{- \\frac{T^{2} g l \\left(2 l - 1\\right) \\left(a - 2 l + 1\\right)}{4 \\left(a^{2} - 4 a l + 2 l\\right)}\\right\\}$"
      ],
      "text/plain": [
       "⎧  2                             ⎫\n",
       "⎪-T ⋅g⋅l⋅(2⋅l - 1)⋅(a - 2⋅l + 1) ⎪\n",
       "⎨────────────────────────────────⎬\n",
       "⎪        ⎛ 2              ⎞      ⎪\n",
       "⎩      4⋅⎝a  - 4⋅a⋅l + 2⋅l⎠      ⎭"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res1 = solveset(res1a, H)\n",
    "res1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "11920e42",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{T^{3} g^{2} l m \\left(2 l - 1\\right) \\left(a - 2 l + 1\\right)^{2}}{16 \\left(a^{2} - 4 a l + 2 l\\right)}$"
      ],
      "text/plain": [
       " 3  2                            2\n",
       "T ⋅g ⋅l⋅m⋅(2⋅l - 1)⋅(a - 2⋅l + 1) \n",
       "──────────────────────────────────\n",
       "         ⎛ 2              ⎞       \n",
       "      16⋅⎝a  - 4⋅a⋅l + 2⋅l⎠       "
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res2a = simplify(res1aa.subs(H, res1.args[0])); factor(res2a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "421a66fb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARkAAAAmCAYAAADwS+/bAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAABJ0AAASdAHeZh94AAAJ2UlEQVR4nO2de7BVVR3HPwiFgg4qTWSCTmZJ1HQ3XDNDYKAZKKuZqGHEnIhqevlKp9ESavrxM3NEA8EGprQmwpixdIqRZHTUlHxkPuJp6hhqKT5GRBgwI6HbH2ttXWef/Thnn/04B9Zn5s69a+31+K3fXnud9djnewcNDAzg8Xg8ZTGkyspU9UjgDlvvEGCpiFzXRv4xwPXAO4F9wI9E5MYSTPV4upZeew4Oqbi+3cAUEQmAjwLzVXVkG/n3AReKyDhgBrBEVYcXb6bH09X01HNQ6UxGRPYD/7bBocAg+9Nq/heAF+zfL6rqduBo4DUAVb0C6BeR6Z3YqapHAY8DE0Vkaxv5Cqm/HVT1RuABEVkUcy1XO6qim/zV7b6CJn8lPgd12ZTk20IGGVWdAdyWkWyuiKy0S6Z1wPuAi0Vke846+4HBIvKsEx0AG/KUF2E+sDbsbKo6D/g8cBKwF3gAmCciWyL5iqofW+8U4CKgH3g38BURWRFJdimwTlV/ISK70tpRFT3qr1p8FcX67nJgmYicF7kcEPFXwnNQl02xvi1quXQPcIzz8yywKBJ3A4CI7BSRPuA9wFmqOqrdylT1aGAl8I3IpQBYn68Jb5Y9DPga8EsneiqwHJgIfBwzXb3D2lFo/REOB7YAFwCvxyUQkc3AU8AX3fiEdlTFVAr0l6quUNUFLdSby19l+KoNm908p2L69KaEJAGOv1Keg8Jox6akvljITEZEXsfeVFUdAYwG7hORF1PyvKSqG4HJwE02bz+wGDgF2Ipp3DpguojcbdMMBVYDV4jI/WF5qvouYBR2VFXV0ZjB7kzgm8CpwJPAbExnXAycjOmUs0TkX7aoTwEDwH2OrZ9wbVfVOcAu4DRgTVz9RSAia4G1tvwVKUlvBr4ALHPimtphy/kBcAZwAuae3QKcbe9hUXb3mr9ifWXLKd1ftp4RwCrgq4DEXI/279jnoE6bLE19sYyN3wmYfZZHYowapapH2L9HAFOAJ2x4LHAX8FdbxiXAbzED4QabZhCwAviTiFwfKT7AdIInbLjP/j4bUMxG81Cb/0rM9PhjwDuA7zjlTAYeEZG0s/0jML57NaX+KnkQOEVVD3PiktoxBOOTD2I6w3TgwpLt63Z/pd3zqvx1LXCTiNyVcD3A+ivjOajFJieuqS+WsfHbD2x3ZgYuxwPXWicNAn5qp1gA1wC3i8h3bfgxVT0Dcxq108adhpmJbFLVmTZuji0jADbbzWVseBcwW0ReAlDV2zEzm7Ei8oqNW4dZzrk2Pp/RxqWYge8vTly0/ip5HngbZh8i3FOIbYeILHCC/1TVW4CxJdvX7f5KvOdV+EtVvw6cSGSZESHA+ktVJ5H8HNRikxPX1BfLGGQmkLDOFpEHrWEN2HP/6THX9gIbnfz3kjz7CmictvUBfwwHGMtxwO/DAcaJe8gJHwa4eaK2LgYmAZMizo3W7+a5DPh+UpmWaeGSMAfh1N2dyTS1w/r5YmAacCzwdszs7sq4QouwO6e/5mNmmiFDgQFVvciJO11E7smwLYmov2LveTv+ymuzqp6E2VSdJCJvpNgcYP2V8RxE7Wr7HuaxyaGpL5Y1yPwhR559NG8ufQC4s8UyAswnphu+JpJmPPDDSFwfZloYsh04Kq4CVb0aMxOaJiJPZdTvsgT4TcK1kLiZX6uEG6ovO3EN7bDvIz0E/BlzAvMcsN/GbUgodwkd2N2Bv34G/M4JLwS20Xg/t2XYlUbUX033PIe/8tocLtkfVdUwbjAwRVW/BQwXkb2k+yuNJbR/DzuxqakvFjrIqOrhwPuBv7WZdT+mEcN4652XfszyaHEL9Q7DHImvt+HhwHtp3IkfCYyJxI0BRtI481oPfDmmjqWYKeo0EXk8rf4o9pg+11F9i3wI2BaZtUXb8WngUMzycQBAVediNsE3xBXaid0d+msHsMNJvxvYISL/yGNLDFF/xd3ztvzVgc2rgYcjcb/CHFJcDvw3y19p5LyHndjU1BeLnsmMx+y1tDvIPIxZGl2lqoswxl9tr21oIf+H7e9NkbCbN7B1PBqJ2wO4HeE2YKGqjnT2bZYBc4CZwKt2Vx1gj4jsiam/EOygfaINHgIcp6oBpvO6nz6TaX5PKdqOVzAPyExV3Qycjpne76ax/UXY3Wv+arrnVOQvu9+4M9KO16zNW2y4FH+VZFNTXyz6dGkCZrM1OjVOxR51z8UcJW4CzsPsnr8cM82OIwCeFJHwbeI+G3bfgBwPbBGRfU5cH7BRRP7n2LIZs0N+ppPuHMwJyZ2YNy3Dn3C9Ha2/KE7GfFKsx6xx1f59aZhAVQ8FPgc0fAcsph1rgZ8Dvwbuxwzkq4BNGSdpeegpfyXc8yr9lUVAOf7qhICITUl9cVA3fgvbnj7dCmwVkXNqqP+TmLXmuJpOP1pGVc8FPisiM2Ku9Uw7qiLJX95XnZPk26q/IBmLqk5S1VmqeoKqfgTziREAl9Vhj4jcinmZaHQd9bfJG8D5cRd6rB1VEesv76tCiPVtV8xkVHUWZjf+WMyu9N2Y77o8V6ddHo+nc7pikPF4PAcuDadLqvoM5u3HVlklImlvBHo8noOc6BH2VuA/beTPev3e4/Ec5PjlksfjKZWiRKv8SOXxeGLxM5kDEO0xoeluxPuwOLriPRlP4fSU0HSX4n1YEH4m0+VoAQLXVoHwM2XqwOZFaxATj9arKWLsTvqu9WG3U+l/K/C8haoOBhZgRIGOwXy3ZxWwIPL9qo4ErrV+oeksAooVE88jYp4mxl6JDw9k/HKpPr4HnAt8G6O0doENzwsTaIcC19odQtNZBBQrJj6VNkXMkwSwbb2l+/BAx89k6mMisEZE1tjwM6p6M0aLOCS3wHU3CE1n2ZkgRN0RHYiYNwlgV+HDgwE/k6mPe4FpVkAdVR2H+eRd66TJJXDdRULTqXZSjZh4qyLmDQLYFfrwgMfPZOpjIeYB+Luq7sfcix+LyHInTV6B6zTB9UJoUWg6y86A8sXEWxUxjwpgl+7DgwU/yNTHbOBLwFkYtb4AWKqqT4tIuAeTS+C6VaHpvELhrQpNtyDEHVCimHibIuYNAtjtiHV70vGDTH1cBfxERG6w4c2qejxm4zccZIoQuE5jCfmEwjOFpjHSlVl2BpQkJp5DxDxOjN1TAH6QqY9hmIfOZT+Nn54dC1yn0YFQ+GoyhKaz7CxTTDyniHmcGLunAPwgUx9rgEtU9WnMcmk85j9ZrnTS1CZwnUaLQtNZdoanaEWLiecVMY8TY/cUgF9z1sf5mP8Bvhx4DFiEEWB+c4+kBwSu08iyM6Accey2RcyTBLA9xeC/VtDleIHr8kkTY/d0jp/JdDle4LoSEsXYPZ3jZzIej6dU/g9RUe0ByKYELQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle - \\frac{T^{3} g^{2} l m \\left(2 l - 1\\right) \\left(a - 2 l + 1\\right) \\left(a + 4 l^{2} - 4 l\\right)}{8 \\left(a^{2} - 4 a l + 2 l\\right)^{2}}$"
      ],
      "text/plain": [
       "  3  2                             ⎛       2      ⎞ \n",
       "-T ⋅g ⋅l⋅m⋅(2⋅l - 1)⋅(a - 2⋅l + 1)⋅⎝a + 4⋅l  - 4⋅l⎠ \n",
       "────────────────────────────────────────────────────\n",
       "                                   2                \n",
       "                 ⎛ 2              ⎞                 \n",
       "               8⋅⎝a  - 4⋅a⋅l + 2⋅l⎠                 "
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res3a = factor(cancel(diff(res2a, a))); res3a\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "c2da3307",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{2 l - 1, - 4 l^{2} + 4 l\\right\\}$"
      ],
      "text/plain": [
       "⎧              2      ⎫\n",
       "⎨2⋅l - 1, - 4⋅l  + 4⋅l⎬\n",
       "⎩                     ⎭"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res4 = solveset(res3a, a); res4.args[0]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "id": "96c0a779",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{\\frac{T^{2} g}{8}\\right\\}$"
      ],
      "text/plain": [
       "⎧ 2  ⎫\n",
       "⎪T ⋅g⎪\n",
       "⎨────⎬\n",
       "⎪ 8  ⎪\n",
       "⎩    ⎭"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(res1.subs(a, res4.args[0].args[1]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "817ab9df",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "name": "python3"
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    "name": "ipython",
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   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "pygments_lexer": "ipython3",
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