{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "e6e99643-fa2f-40c5-b748-19ae6baae4f1",
   "metadata": {},
   "source": [
    "# Inverting Kepler's equation in ODEs\n",
    "\n",
    "In the [previous tutorial](<./Comparing coordinate systems.ipynb>) we considered the numerical solution of the Stark problem, whose Hamiltonian, in Cartesian coordinates, reads\n",
    "\n",
    "$$\n",
    "\\mathcal{H}_\\mathrm{cart}\\left(v_x, v_y, v_z, x, y, z \\right) = \\frac{1}{2}\\left( v_x^2+v_y^2+v_z^2 \\right) - \\frac{1}{\\sqrt{x^2+y^2+z^2}} - \\varepsilon z.\n",
    "$$\n",
    "\n",
    "We saw how a reformulation of the Stark problem using Delaunay elements, instead of Cartesian coordinates, can help reducing the total number of timesteps per unit of integration time. The Delaunay elements are the Hamiltonian version of the [Keplerian orbital elements](https://en.wikipedia.org/wiki/Orbital_elements), to which they are related via the following equations (valid in adimensional units):\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "L & = \\sqrt{a}, & l & =  M, \\\\\n",
    "G & = \\sqrt{a\\left( 1 - e^2\\right)}, & g & = \\omega, \\\\\n",
    "H & = \\sqrt{a\\left( 1 - e^2\\right)} \\cos i, & h & =  \\Omega.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "The Hamiltonian of the Stark problem in Delaunay elements reads:\n",
    "\n",
    "$$\n",
    "\\mathcal{H}_\\mathrm{Del} \\left( L, G, H, l, g, h \\right) = -\\frac{1}{2L^2}-\\varepsilon L\\sqrt{1-\\frac{H^2}{G^2}}\\left[ L\\left( \\cos E - \\sqrt{1-\\frac{G^2}{L^2}} \\right)\\sin g + G\\sin E \\cos g \\right].\n",
    "$$\n",
    "\n",
    "In this Hamiltonian, $E$ is the eccentric anomaly, which is a function of $L$, $G$ and $l$ implicitly defined by [Kepler's equation](https://en.wikipedia.org/wiki/Kepler%27s_equation):\n",
    "\n",
    "$$\n",
    "l = E - \\sqrt{1-\\frac{G^2}{L^2}} \\sin E.\n",
    "$$\n",
    "\n",
    "Because Kepler's equation is trascendental, it cannot be inverted to yield an explicit expression for $E$ in terms of elementary functions. In the [previous tutorial](<./Comparing coordinate systems.ipynb>) we worked around this issue by leaving $E$ as an unspecified function in the Hamiltonian; we then augmented Hamilton's equations via the derivatives of $E$ (which can be formulated explicitly) and numerically solved the augmented system of differential equations.\n",
    "\n",
    "Here we will employ a different approach, and we will use instead the function ``kepE()`` provided by heyoka.py's expression system. As the name suggests, ``kepE()`` represents symbolically the inversion of Kepler's elliptic equation. That is, ``kepE(e, M)`` is the bivariate function $E\\left(e,M\\right)$ implicitly defined by Kepler's elliptic equation:\n",
    "\n",
    "$$\n",
    "M = E - e \\sin E.\n",
    "$$\n",
    "\n",
    "For the evaluation of ``kepE(e, M)`` heyoka.py uses an iterative numerical scheme. The high-order derivatives necessary to implement Taylor's integration method are built using automatic-differentiation techniques starting from the evaluation of ``kepE(e, M)``.\n",
    "\n",
    "Let us begin by writing down the Hamiltonian:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "c82ace35-08ea-4c28-b04c-81b468c5193b",
   "metadata": {},
   "outputs": [],
   "source": [
    "import heyoka as hy\n",
    "\n",
    "# Numerical value for the eps constant.\n",
    "eps = 1e-3\n",
    "\n",
    "# Symbolic variables for the Delaunay arguments.\n",
    "L, G, H, l, g, h = hy.make_vars(\"L\", \"G\", \"H\", \"l\", \"g\", \"h\")\n",
    "\n",
    "# Define E as the solution to Kepler's equation.\n",
    "E = hy.kepE(hy.sqrt(1 - G**2 * L**-2), l)\n",
    "\n",
    "# The Hamiltonian.\n",
    "Ham_del = -0.5 * L**-2 - eps * L * hy.sqrt(1.0 - H**2 * G**-2) * (\n",
    "    L * (hy.cos(E) - hy.sqrt(1.0 - G**2 * L**-2)) * hy.sin(g)\n",
    "    + G * hy.sin(E) * hy.cos(g)\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6a91824-ca5b-4140-8485-2434733ad532",
   "metadata": {},
   "source": [
    "Note how, in terms of Delaunay arguments, $e=\\sqrt{1-\\frac{G^2}{L^2}}$ and $M=l$.\n",
    "\n",
    "Unlike in the [previous tutorial](<./Comparing coordinate systems.ipynb>), and thanks to the fact that ``kepE()`` supports symbolic differentiation, we can now use the standard formulation of Hamilton's equations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a1b17f85-4efc-4ec0-8b20-fb1aa8a3bddb",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define the system of ODEs.\n",
    "sys = [\n",
    "    (L, -hy.diff(Ham_del, l)),\n",
    "    (G, -hy.diff(Ham_del, g)),\n",
    "    (H, -hy.diff(Ham_del, h)),\n",
    "    (l, hy.diff(Ham_del, L)),\n",
    "    (g, hy.diff(Ham_del, G)),\n",
    "    (h, hy.diff(Ham_del, H)),\n",
    "]"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72fe3a74-e035-440c-b420-3646aabb4516",
   "metadata": {},
   "source": [
    "We can now proceed to the creation of the integrator object. The initial conditions are taken from the [previous tutorial](<./Comparing coordinate systems.ipynb>):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "56aedf2a-767a-4333-a202-778a99c8430f",
   "metadata": {},
   "outputs": [],
   "source": [
    "ta = hy.taylor_adaptive(\n",
    "    sys,\n",
    "    [\n",
    "        1.0045488165591647,\n",
    "        0.9731906288081488,\n",
    "        -0.9683287292736491,\n",
    "        2.776991035843252,\n",
    "        4.314274521695855,\n",
    "        3.3415926535897924,\n",
    "    ],\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5cfa5b10-aec7-41d6-bb2f-4a31ac51c6a3",
   "metadata": {},
   "source": [
    "Let us now integrate up to $t=250$ on a time grid. As usual, we will provide a callback to periodically reduce the $l$ angle modulo $2\\pi$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b0c52e16-f32c-49d4-9bb3-0cd07e7ae1af",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "\n",
    "\n",
    "# Callback to reduce l to the\n",
    "# [-pi, pi] range.\n",
    "def mod_cb_del(ta):\n",
    "    l = ta.state[3]\n",
    "    if l < -np.pi or l > np.pi:\n",
    "        ta.state[3] = (l + np.pi) % (2 * np.pi) - np.pi\n",
    "\n",
    "    return True\n",
    "\n",
    "\n",
    "# Define a time grid for the integration.\n",
    "t_grid = np.linspace(0, 250, 1000)\n",
    "\n",
    "# Integrate.\n",
    "_, _, _, _, _, out_del = ta.propagate_grid(t_grid, callback=mod_cb_del)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "26208487-5c97-4412-9f6a-9efad835a9c2",
   "metadata": {},
   "source": [
    "Let us take a look at the time evolution of the Delaunay elements:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "77b9c512-ad7d-43ab-b11b-9a15b00d6d28",
   "metadata": {},
   "outputs": [
    {
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      "text/plain": [
       "<Figure size 1200x600 with 6 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib.pylab import plt\n",
    "\n",
    "\n",
    "def plot_t_evol(out, labels):\n",
    "    fig = plt.figure(figsize=(12, 6))\n",
    "\n",
    "    max_abs = 8\n",
    "\n",
    "    ncoord = out.shape[1]\n",
    "    ncols = 3\n",
    "    nrows = ncoord // ncols + (ncoord % ncols)\n",
    "\n",
    "    for i in range(0, ncoord):\n",
    "        ax = fig.add_subplot(nrows, ncols, i + 1)\n",
    "        ax.plot(t_grid, out[:, i])\n",
    "        ax.set_xlabel(\"Time\")\n",
    "        ax.set_ylabel(labels[i])\n",
    "        ax.set_ylim(-max_abs, max_abs)\n",
    "\n",
    "    plt.tight_layout()\n",
    "\n",
    "\n",
    "plot_t_evol(out_del, [\"$L$\", \"$G$\", \"$H$\", \"$l$\", \"$g$\", \"$h$\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6f5585c7-2429-4f26-9cc8-8e8269e254d5",
   "metadata": {},
   "source": [
    "It can be confirmed by direct numerical comparison that the integration of the Stark problem using ``kepE()`` produced results matching those from the [previous tutorial](<./Comparing coordinate systems.ipynb>) to machine precision."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}