{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a0d44cc6",
   "metadata": {},
   "outputs": [],
   "source": [
    "# ] add Elliptic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "92d7682d",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Elliptic\n",
    "using PyPlot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8cb74d27",
   "metadata": {},
   "outputs": [],
   "source": [
    "\"\"\"\n",
    "Analytic solution of the pendulum ode\n",
    "\n",
    "theta'' + sin(theta) = 0\n",
    "\n",
    "theta(0) = 0, max(theta) = thetamax;\n",
    "theta'(0) = sqrt(2*(1 - cos(thetamax))) - from the conservation of energy\n",
    "\"\"\"\n",
    "function pendulum_analytics(t, thetamax)\n",
    "    m = sin(thetamax / 2.)\n",
    "    th = 2 * asin(m * Jacobi.sn(t, m))\n",
    "    thd = 2 * m * Jacobi.cn(t, m)\n",
    "    (th, thd)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c5bd6343",
   "metadata": {},
   "outputs": [],
   "source": [
    "? pendulum_analytics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "10a50a66",
   "metadata": {},
   "outputs": [],
   "source": [
    "tmax = 4*pi\n",
    "np = 100\n",
    "t = range(0.0, tmax, length=np)\n",
    "thetamax = pi/10\n",
    "\n",
    "theta = zeros(np)\n",
    "thetadot = zeros(np)\n",
    "\n",
    "theta[1] = 0.0\n",
    "thetadot[1] = sqrt(2*(1.0 - cos(thetamax)))\n",
    "\n",
    "for i = 2:np\n",
    "    theta[i], thetadot[i]  = pendulum_analytics(t[i], thetamax)\n",
    "end\n",
    "\n",
    "plot(t, theta, label=L\"\\theta\")\n",
    "plot(t, thetadot, label=L\"\\dot{\\theta}\")\n",
    "grid(true)\n",
    "legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bef9bc8f",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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