{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Recently, I've come across the term Verlet integration several times: an [interesting numerical exploration of a moon explosion by Jason Cole](https://jasmcole.com/2017/09/20/the-moon-blew-up-without-warning-and-for-no-apparent-reason/) and the [wonderful Lennard-Jones molecular dynamics simulator by Daniel Schroeder](http://physics.weber.edu/schroeder/md/). I've been interested in numerical simulation for a long time (and wrote about it [here](http://flothesof.github.io/harmonic-oscillator-three-methods-solution.html) regarding the harmonic oscillator as well as [here](http://flothesof.github.io/charged-particle-trajectories-E-and-B-fields.html) regarding the movement of charged particles in E and B fields) so I think it's time to explore what exactly this numerical ODE integration technique is."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Verlet integration "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to [Wikipedia](https://en.wikipedia.org/wiki/Verlet_integration), Verlet integration is named after [Loup Verlet](https://en.wikipedia.org/wiki/Loup_Verlet), a French physicist who published [a paper in 1967 about a model of the Argon gas using 864 particles](https://journals.aps.org/pr/abstract/10.1103/PhysRev.159.98) following forces obeying the Lennard-Jones potential. The integration method that he used in the original paper is written as follows:\n",
    "\n",
    "$$\n",
    "\\vec{r}_i (t+h) = -\\vec{r}_i(t-h)  + 2 \\vec{r}_i (t) + \\sum_{j \\neq i} \\vec{f}(r_{ij}(t)) h^2\n",
    "$$\n",
    "\n",
    "For our purpose, we will use the formulation found in Wikipedia:\n",
    "$$\n",
    "\\vec{r}_i (t+h) = -\\vec{r}_i(t-h)  + 2 \\vec{r}_i (t) + \\vec{a}(r_{i}(t)) h^2\n",
    "$$\n",
    "\n",
    "In this equation, $\\vec{r}$ is the position vector for the particle we are modeling and $\\vec{a}$ is its acceleartion.\n",
    "\n",
    "As stated in the Wikipedia article, the interesting thing about this integration scheme is that it is an order more accurate than integration by simple Taylor expansion alone and that is does not use the velocity vector at all. For me, this is quite surprising since most numerical integration techniques I have used before integrate acceleration to get velocity and integrate velocity to get positions.\n",
    "\n",
    "This is all the theory we need. As we will see in the remainder of this post, this formulation is very general and will allow us to play with many examples: oscillators, particle systems, pendulums and many more."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Applications"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Harmonic oscillator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's start with the harmonic oscillator since it is a good scalar example. The harmonic oscillator's  equation is: \n",
    "$$\n",
    "\\ddot{x} + \\omega_0^2 x = 0\n",
    "$$\n",
    "\n",
    "Therefore, the scalar acceleration $a$ is computed as $-\\omega_0^2 x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "class VerletSolver:\n",
    "    def __init__(self, dt, x0, xd0, acc_func):\n",
    "        \"Inits the solver.\"\n",
    "        self.dt = dt\n",
    "        self.dt_squared = dt**2\n",
    "        self.t = dt\n",
    "        self.acc_func = acc_func\n",
    "        self.x0 = x0\n",
    "        self.xd0 = xd0\n",
    "        # the second position vector is obtained by double integration of acceleration\n",
    "        # using the initial velocity xd0\n",
    "        x1 = acc_func(x0) * dt**2 / 2. + xd0 * dt + x0\n",
    "        self.x = [x1, x0]\n",
    "        \n",
    "    def step(self):\n",
    "        \"Steps the solver.\"\n",
    "        xt, xtm1 = self.x\n",
    "        xtp1 = - xtm1 + 2 * xt + self.acc_func(xt) * self.dt_squared\n",
    "        self.x = (xtp1, xt)\n",
    "        self.t += self.dt\n",
    "        \n",
    "    def step_until(self, tmax, snapshot_dt):\n",
    "        \"Steps the solver until a given time, returns snapshots.\"\n",
    "        ts = [self.t]\n",
    "        vals = [self.x[0]]\n",
    "        niter = max(1, int(snapshot_dt // self.dt))\n",
    "        while self.t < tmax:\n",
    "            for _ in range(niter):\n",
    "                self.step()\n",
    "            vals.append(self.x[0])\n",
    "            ts.append(self.t)\n",
    "        return np.array(ts), np.array(vals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_harmonic(x):\n",
    "    \"\"\"Acceleration for harmonic oscilator.\"\"\"\n",
    "    omega_0 = 1.\n",
    "    return -omega_0 * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "solver = VerletSolver(0.01, 2, -0.01, acc_func_harmonic)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ts, vals = solver.step_until(12, snapshot_dt=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(ts, vals, '-o')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see how \"stable\" this is if we vary the integration step size."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for step_size in [0.001, 0.01, 0.05, 0.25]:\n",
    "    solver = VerletSolver(step_size, 2, -0.01, acc_func=acc_func_harmonic)\n",
    "    ts, vals = solver.step_until(12, snapshot_dt=0.5)\n",
    "    plt.plot(ts, vals, \"-o\", label='step_size: {}'.format(step_size))\n",
    "plt.legend()\n",
    "plt.ylim(-2.1, 2.1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As the above graph shows, the Verlet integration works very well in this case."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A two-dimensional spring "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's move on to a two-dimensional spring. The difference with the previous example is that the spring is described by a vector instead of a scalar: its position depends on two coordinates. This integrates easily with the vector Verlet update equation that was explained in the theory section.\n",
    "\n",
    "This time, we thus use a 2d NumPy array to store initial positions and velocities `x0`, `xd0`.\n",
    "\n",
    "One more thing we need is the description of the behavior of a spring. To quote H-P Langtangen:\n",
    "\n",
    "> Stretching the elastic wire a distance ΔL gives rise to a spring force kΔL in the opposite direction of the stretching.\n",
    "\n",
    "This is what we implement below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_spring(x):\n",
    "    \"\"\"Acceleration for a 2D spring.\"\"\"\n",
    "    k = 10\n",
    "    stretch = np.linalg.norm(x) \n",
    "    n = x / np.linalg.norm(x)\n",
    "    acc = -  k * stretch * n\n",
    "    return acc"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's define initial conditions with a position and a speed and then run the simulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.array([0, -1], dtype=np.float)\n",
    "xd0 = np.array([0.0, 0.1], dtype=np.float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "spring = VerletSolver(0.01, x0, xd0, acc_func_spring)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ts, vals = spring.step_until(12, snapshot_dt=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(vals[:, 0], vals[:, 1])\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the trajectory according to time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(ts, vals[:, 0])\n",
    "plt.plot(ts, vals[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or as a frame-by-frame animation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "from IPython.display import HTML\n",
    "import matplotlib.animation as mpl_anim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    ax.plot(vals[index, 0], vals[index, 1], 'o')\n",
    "    ax.plot(vals[index-5:index+1, 0], vals[index-5:index+1, 1], '-')\n",
    "    ax.set_xlim(-1, 1)\n",
    "    ax.set_ylim(-1, 1)\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we give the spring a slight nudge to the right, we can get elliptical oscillations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xd0 = np.array([-0.5, .1], dtype=np.float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "spring = VerletSolver(0.1, x0, xd0, acc_func_spring)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ts, vals = spring.step_until(12, snapshot_dt=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(vals[:, 0], vals[:, 1])\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the trajectory according to time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(ts, vals[:, 0])\n",
    "plt.plot(ts, vals[:, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    ax.plot(vals[index, 0], vals[index, 1], 'o')\n",
    "    ax.plot(vals[index-5:index+1, 0], vals[index-5:index+1, 1], '-')\n",
    "    ax.set_xlim(-1, 1)\n",
    "    ax.set_ylim(-1, 1)\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Shooting a cannonball "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_cannonball(x):\n",
    "    \"\"\"Force function for a cannonball.\"\"\"\n",
    "    g = 10.\n",
    "    acc = g * np.array([0., -1.])\n",
    "    return acc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.array([0, 0], dtype=np.float)\n",
    "xd0 = np.array([10.0, 10.0], dtype=np.float)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "cannonball = VerletSolver(0.001, x0, xd0, acc_func_cannonball)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ts, vals = cannonball.step_until(2, snapshot_dt=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(vals[:, 0], vals[:, 1])\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the trajectory according to time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(ts, vals[:, 0], label='x')\n",
    "plt.plot(ts, vals[:, 1], label='y')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now fire a lot of cannonbals."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "for angle in np.deg2rad(np.linspace(0, 90, num=20)):\n",
    "    x0 = np.array([0, 0], dtype=np.float)\n",
    "    xd0 = 15 * np.array([np.cos(angle), np.sin(angle)], dtype=np.float)\n",
    "    cannonball = VerletSolver(0.001, x0, xd0, acc_func_cannonball)\n",
    "    ts, vals = cannonball.step_until(2, snapshot_dt=0.1)\n",
    "    plt.plot(vals[:, 0], vals[:, 1], '-ok')\n",
    "plt.axis('equal')\n",
    "plt.axis([0, 20, 0, 7])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's do an animation of cannonball isochrones. We first precompute the trajectories:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t_max = 3.5\n",
    "trajectories = {}\n",
    "angles = np.deg2rad(np.linspace(25, 85, num=20))\n",
    "for angle in angles:\n",
    "    x0 = np.array([0, 0], dtype=np.float)\n",
    "    xd0 = 15 * np.array([np.cos(angle), np.sin(angle)], dtype=np.float)\n",
    "    cannonball = VerletSolver(0.001, x0, xd0, acc_func_cannonball)\n",
    "    ts, vals = cannonball.step_until(t_max, snapshot_dt=0.01)\n",
    "    trajectories[angle] = (ts, vals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And then animate the plot of the simultaneous trajectories as a function of time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    ax.clear() \n",
    "    for angle in angles:\n",
    "        ts, vals = trajectories[angle]\n",
    "        index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "        ax.plot(vals[:index, 0], vals[:index, 1], '-k')\n",
    "        ax.plot(vals[index, 0], vals[index, 1], 'ok')\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "    ax.set_xlim(-1, 19)\n",
    "    ax.set_ylim(0, 20)\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Asteroids around the earth "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also simulate some asteroids orbiting around the earth (considered fixed).\n",
    "\n",
    "First, we generate a set of asteroids as well as their velocities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_asteroids(n, rmin, rmax):\n",
    "    \"\"\"Creates a bunch of asteroids.\"\"\"\n",
    "    rs = np.random.uniform(rmin, rmax, size=n) \n",
    "    thetas = np.random.uniform(0, 360, size=n)\n",
    "    x = np.stack([rs * np.cos(thetas), rs * np.sin(thetas)], axis=1)\n",
    "    n = np.stack([rs * np.sin(thetas), -rs * np.cos(thetas)], axis=1)\n",
    "    xd = n \n",
    "    return x, xd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0, xd0 = make_asteroids(200, 0.5, 1.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "ax.quiver(x0[:, 0], x0[:, 1], xd0[:, 0], xd0[:, 1])\n",
    "ax.plot([0], [0], 'ro', ms=10, label='earth')\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then create a plotting function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_asteroids(x, ax, **plot_kwargs):\n",
    "    xy = x.reshape(-1, 2)\n",
    "    ax.plot(xy[:, 0], xy[:, 1], 'o', **plot_kwargs)\n",
    "    ax.plot([0], [0], 'ro', ms=10, label='earth')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "plot_asteroids(x0, ax, ms=4, alpha=0.5)\n",
    "ax.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now define the gravitational force, which always pulls the asteroids towards the earth."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_asteroids(x):\n",
    "    \"\"\"Acceleration for asteroids orbiting the earth.\"\"\"\n",
    "    xy = x.reshape(-1, 2)\n",
    "    r = np.linalg.norm(xy - np.array([0., 0.]), axis=1)\n",
    "    n = xy / r[:, np.newaxis]\n",
    "    acc = - 0.0000001 * 1/r[:, np.newaxis]**2  * n\n",
    "    return acc.ravel()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "asteroid_field = VerletSolver(1., \n",
    "                              x0.ravel(), \n",
    "                              (xd0.ravel() + np.random.randn(xd0.size) / 20) / 2000, \n",
    "                              acc_func_asteroids)\n",
    "ts, vals = asteroid_field.step_until(40000, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 100\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_asteroids(vals[index], ax)\n",
    "    if index>4:\n",
    "        plot_asteroids(vals[index-1], ax, alpha=0.5, ms=5)\n",
    "        plot_asteroids(vals[index-2], ax, alpha=0.4, ms=4)\n",
    "        plot_asteroids(vals[index-3], ax, alpha=0.3, ms=3)\n",
    "        plot_asteroids(vals[index-4], ax, alpha=0.2, ms=2)\n",
    "    ax.axis([-2.5, 2.5, -2.5, 2.5])\n",
    "    ax.legend(loc='upper right')\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=200, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's also render a close-up view:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 100\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_asteroids(vals[index], ax)\n",
    "    if index>4:\n",
    "        plot_asteroids(vals[index-1], ax, alpha=0.5, ms=5)\n",
    "        plot_asteroids(vals[index-2], ax, alpha=0.4, ms=4)\n",
    "        plot_asteroids(vals[index-3], ax, alpha=0.3, ms=3)\n",
    "        plot_asteroids(vals[index-4], ax, alpha=0.2, ms=2)\n",
    "    ax.axis([-1.25, 1.25, -1.25, 1.25])\n",
    "    ax.legend(loc='upper right')\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=200, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As one can see, it seems that particles closer to the earth have faster orbits, while particles more farther away go more slowly."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A pendulum "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now turn to pendulums. A pendulum is basically a spring with an additional force: the pull of gravity along the vertical axis. Let's write that as a function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_pendulum(x):\n",
    "    \"\"\"Force function with spring of fixed size.\"\"\"\n",
    "    g = 1.\n",
    "    l0 = 1.\n",
    "    elongation = np.linalg.norm(x) - l0\n",
    "    k = 100.\n",
    "    acc = - k * elongation * x / np.linalg.norm(x) + g * np.array([0, -1])\n",
    "    return acc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.array([1, 0], dtype=np.float)\n",
    "xd0 = np.array([0.0, 0.0], dtype=np.float)\n",
    "pendulum = VerletSolver(0.01, x0, xd0, acc_func_pendulum)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "ts, vals = pendulum.step_until(15, snapshot_dt=0.05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(vals[:, 0], vals[:, 1])\n",
    "plt.axis('equal');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the trajectory according to time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(ts, vals[:, 0], label='x')\n",
    "plt.plot(ts, vals[:, 1], label='y')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or as an animation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    ax.plot([0, vals[index, 0]], [0, vals[index, 1]])\n",
    "    ax.plot(vals[index, 0], vals[index, 1], 'o')\n",
    "    ax.plot(vals[index-10:index+1, 0], vals[index-10:index+1, 1], '-')\n",
    "    ax.axis('equal')\n",
    "    ax.axis([-1.5, 1.5, -2, 1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One interesting thing about pendulums is that when you make the spring less stiff, you can obtain more varied behaviours:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_pendulum_elastic(x):\n",
    "    \"\"\"Force function with spring of fixed size.\"\"\"\n",
    "    g = 1.\n",
    "    l0 = 1.\n",
    "    elongation = np.linalg.norm(x) - l0\n",
    "    k = 5.\n",
    "    acc = - k * elongation * x / np.linalg.norm(x) + g * np.array([0, -1])\n",
    "    return acc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.array([1, 0], dtype=np.float)\n",
    "xd0 = np.array([0.0, 0.0], dtype=np.float)\n",
    "\n",
    "pendulum = VerletSolver(0.01, x0, xd0, acc_func_pendulum_elastic)\n",
    "\n",
    "ts, vals = pendulum.step_until(30, snapshot_dt=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    ax.plot([0, vals[index, 0]], [0, vals[index, 1]])\n",
    "    ax.plot(vals[index, 0], vals[index, 1], 'o')\n",
    "    ax.plot(vals[index-10:index+1, 0], vals[index-10:index+1, 1], '-')\n",
    "    ax.axis('equal')\n",
    "    ax.axis([-1.5, 1.5, -2, 1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Two particle systems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another interesting example consists of two particles attached by a spring."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_two_spring(x):\n",
    "    \"\"\"Force function with spring of fixed size.\"\"\"\n",
    "    g = 1.\n",
    "    l0 = 1.\n",
    "    x1 = x[:2]\n",
    "    x2 = x[2:]\n",
    "    elongation1 = np.linalg.norm(x1 - x2) - l0\n",
    "    elongation2 = np.linalg.norm(x2 - x1) - l0\n",
    "    k = 10.\n",
    "    acc = np.zeros_like(x)\n",
    "    acc[:2] = - k * elongation1 * (x1 - x2) / np.linalg.norm(x1 - x2)\n",
    "    acc[2:] = - k * elongation2 * (x2 - x1) / np.linalg.norm(x2 - x1)\n",
    "    return acc"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.array([0.75, 0, -0.75, 0], dtype=np.float)\n",
    "xd0 = np.array([0.0, 0.0, 0.0, 0.0], dtype=np.float)\n",
    "twosprings = VerletSolver(0.01, x0, xd0, acc_func_two_spring)\n",
    "ts, vals = twosprings.step_until(2.8, snapshot_dt=0.05)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot the trajectory according to time:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(ts, vals[:, 0], label='x')\n",
    "plt.plot(ts, vals[:, 1], label='y')\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's animate this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    ax.plot(vals[index, [0, 2]], vals[index, [1, 3]], 'o', ms=50)\n",
    "    ax.axis('equal')\n",
    "    ax.axis([-1.5, 1.5, -1, 1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happens if the particles start with a little spin?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.array([0.75, 0, -0.75, 0], dtype=np.float)\n",
    "xd0 = np.array([0.0, 0.1, 0.0, -.1], dtype=np.float)\n",
    "twosprings = VerletSolver(0.01, x0, xd0, acc_func_two_spring)\n",
    "ts, vals = twosprings.step_until(8, snapshot_dt=0.05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    ax.plot(vals[index, [0, 2]], vals[index, [1, 3]], 'o')\n",
    "    # tail 1\n",
    "    ax.plot(vals[0:index+1, 0], vals[0:index+1, 1], '-')\n",
    "    # tail 2\n",
    "    ax.plot(vals[0:index+1, 2], vals[0:index+1, 3], '-')\n",
    "    ax.axis('equal')\n",
    "    ax.axis([-1.5, 1.5, -1, 1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They still pull on each other, however the two particles now also \"dance in a circle\" following their initial momentum."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## A chain of oscillators strung together "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's build a chain of oscillators that are fixed on both ends."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def build_chain(n_free):\n",
    "    \"\"\"Returns x0 and xd0 for a chain of n_free horizontal particles.\"\"\"\n",
    "    l0 = 1\n",
    "    xs = np.arange(n_free + 2, dtype=np.float) * l0\n",
    "    ys = np.zeros_like(xs)\n",
    "    x0 = np.c_[xs.reshape(-1, 1), ys.reshape(-1, 1)].ravel()\n",
    "    xd0 = np.zeros_like(x0) \n",
    "    return x0, xd0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib.collections import LineCollection\n",
    "\n",
    "def plot_chain(ax, x):\n",
    "    xy = x.reshape(-1, 2)\n",
    "    line = LineCollection([[start, stop] for start, stop in zip(xy[:-1,:], xy[1:,:])], linestyles='-.')\n",
    "    ax.add_artist(line)\n",
    "    ax.plot(xy[:, 0], xy[:, 1], 'o')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0, xd0 = build_chain(3)\n",
    "x0[4] += 50 / 100"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots()\n",
    "plot_chain(ax, x0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, let's write the acceleration function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_chain(x):\n",
    "    \"\"\"Acceleration of a chain of oscillators strung together. \n",
    "    Node 0 and -1 are assumed fixed.\"\"\"\n",
    "    k = 10.\n",
    "    l0 = 1\n",
    "    xy = x.reshape(-1, 2)\n",
    "    acc = np.zeros_like(xy)\n",
    "    for index in range(1, xy.shape[0] - 1):\n",
    "        x0 = xy[index-1, :]\n",
    "        x1 = xy[index, :]\n",
    "        x2 = xy[index+1, :]\n",
    "        \n",
    "        elongation1 = (np.linalg.norm(x1 - x0) - l0) / l0\n",
    "        elongation2 = (np.linalg.norm(x2 - x1) - l0) / l0\n",
    "        acc[index, :] = - k * elongation1 * (x1 - x0) / np.linalg.norm(x1 - x0) \\\n",
    "                        - k * elongation2 * (x1 - x2) / np.linalg.norm(x2 - x1)\n",
    "    \n",
    "    return acc.ravel()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now let's test this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "chain = VerletSolver(0.01, x0, xd0, acc_func_chain)\n",
    "ts, vals = chain.step_until(5, 0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see what the trajectories of the individual particles are like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots()\n",
    "reshaped_vals = vals.reshape(-1, 5, 2)\n",
    "for particle_index in range(reshaped_vals.shape[1]):\n",
    "    traj = reshaped_vals[:, particle_index, :]\n",
    "    ax.plot(traj[:, 0])\n",
    "    ax.plot(traj[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's animate this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_chain(ax, vals[index])\n",
    "    ax.axis('equal')\n",
    "    ax.axis([-0.5, 4.5, -0.1, 0.1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if one of the extremities is loose? This is just a matter of updating the acceleration of the last particle in the chain with the pulling of its neighbor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def acc_func_chain2(x):\n",
    "    \"\"\"Acceleration of a chain of oscillators strung together. \n",
    "    Node 0 and -1 are assumed fixed.\"\"\"\n",
    "    k = 10.\n",
    "    l0 = 1\n",
    "    xy = x.reshape(-1, 2)\n",
    "    acc = np.zeros_like(xy)\n",
    "    for index in range(1, xy.shape[0] - 1):\n",
    "        x0 = xy[index-1, :]\n",
    "        x1 = xy[index, :]\n",
    "        x2 = xy[index+1, :]\n",
    "        \n",
    "        elongation1 = (np.linalg.norm(x1 - x0) - l0) / l0\n",
    "        elongation2 = (np.linalg.norm(x2 - x1) - l0) / l0\n",
    "        acc[index, :] = - k * elongation1 * (x1 - x0) / np.linalg.norm(x1 - x0) \\\n",
    "                        - k * elongation2 * (x1 - x2) / np.linalg.norm(x2 - x1)\n",
    "    index += 1\n",
    "    x0 = xy[index-1, :]\n",
    "    x1 = xy[index, :]\n",
    "    elongation1 = (np.linalg.norm(x1 - x0) - l0) / l0\n",
    "    acc[index, :] = - k * elongation1 * (x1 - x0) / np.linalg.norm(x1 - x0) \\\n",
    "    \n",
    "    return acc.ravel()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0, xd0 = build_chain(3)\n",
    "x0[4] += 50 / 100\n",
    "chain = VerletSolver(0.01, x0, xd0, acc_func_chain2)\n",
    "ts, vals = chain.step_until(5, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_chain(ax, vals[index])\n",
    "    ax.axis('equal')\n",
    "    ax.axis([-0.5, 4.5, -0.1, 0.1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if we add a little bit of noise to this chain of oscilators?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "xd1 = xd0.copy()\n",
    "xd1[2:] += np.random.randn(8) / 100\n",
    "chain = VerletSolver(0.01, x0, xd1, acc_func_chain2)\n",
    "ts, vals = chain.step_until(40, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_chain(ax, vals[index])\n",
    "    ax.axis('equal')\n",
    "    ax.axis([-4.5, 4.5, -0.1, 0.1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We get a behaviour that resembles that of a N-link pendulum!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if we use more than 5 particles?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0, xd0 = build_chain(28)\n",
    "x0[-2] += .5\n",
    "chain = VerletSolver(0.01, x0, xd0, acc_func_chain2)\n",
    "ts, vals = chain.step_until(40, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(10, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_chain(ax, vals[index])\n",
    "    ax.axis([-0.5, 30.5, -0.1, 0.1])\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compression waves appear! Let's plot this data in the \"earth science style\" to see the wave advancing across time and space at the same time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 8))\n",
    "reshaped_vals = vals.reshape(-1, 30, 2)\n",
    "for time_index in range(0, reshaped_vals.shape[0], 10):\n",
    "    delta_traj = reshaped_vals[time_index, :, 0] - np.arange(30)\n",
    "    ax.plot(delta_traj - time_index / 40, 'k')\n",
    "ax.axis('off');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What we see in the above graph is a double reflexion on the left side of the particle chain, as well as the physical phenomenon of dispersion: the wave started as a sharp peak but ended up as a diffuse velocity field."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2D waves in a lattice "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can of course extend this sort of chaining to a grid. Let's first write a function that creates a grid of particles."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "def build_grid(nx, ny):\n",
    "    \"\"\"Builds a grid of nx times ny nodes.\"\"\"\n",
    "    x = np.arange(nx, dtype=np.float)\n",
    "    y = np.arange(ny, dtype=np.float)\n",
    "    X, Y = np.meshgrid(x, y, indexing='ij')\n",
    "    return X, Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numba import njit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit\n",
    "def grid2linear(X, Y):\n",
    "    \"\"\"Returns a linear vector from a X,Y displacement representation.\"\"\"\n",
    "    return np.concatenate((X.ravel(), Y.ravel()))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit\n",
    "def linear2grid(linear_repr, nx, ny):\n",
    "    \"\"\"Returns a grid X,Y displacement from a linear vector.\"\"\"\n",
    "    n = nx * ny\n",
    "    X, Y = linear_repr[:n], linear_repr[n:2*n]\n",
    "    X = X.reshape((nx, ny))\n",
    "    Y = Y.reshape((nx, ny))\n",
    "    return X, Y"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's test these functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx, ny = 3, 2\n",
    "X, Y = build_grid(nx, ny)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  0.]\n",
      " [ 1.  1.]\n",
      " [ 2.  2.]]\n",
      "[[ 0.  1.]\n",
      " [ 0.  1.]\n",
      " [ 0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "print(X)\n",
    "print(Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  0.,  1.,  1.,  2.,  2.,  0.,  1.,  0.,  1.,  0.,  1.])"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linear_repr = grid2linear(X, Y)\n",
    "linear_repr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([[ 0.,  0.],\n",
       "        [ 1.,  1.],\n",
       "        [ 2.,  2.]]), array([[ 0.,  1.],\n",
       "        [ 0.,  1.],\n",
       "        [ 0.,  1.]]))"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linear2grid(linear_repr, nx, ny)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now code the acceleration function, which will connect each node with its neighbors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [],
   "source": [
    "@njit\n",
    "def acc_func_grid_template(x, nx, ny, k):\n",
    "    \"\"\"Acceleration of a grid of oscillators strung together to its 4 neighbors.\"\"\"\n",
    "    l0 = 1.\n",
    "    l1 = np.sqrt(2)\n",
    "    X, Y = linear2grid(x, nx, ny)\n",
    "    acc_X, acc_Y = np.zeros_like(X), np.zeros_like(Y)\n",
    "    for i in range(nx):\n",
    "        for j in range(ny):\n",
    "            x_center = np.array([X[i, j], Y[i, j]])\n",
    "            for neighbor in [(i+1, j, l0), (i-1, j, l0),\n",
    "                             (i, j+1, l0), (i, j-1, l0),\n",
    "                             (i+1, j+1, l1), (i+1, j-1, l1),\n",
    "                             (i-1, j+1, l1), (i-1, j-1, l1)]:\n",
    "                ii, jj, l = neighbor\n",
    "                if (ii >= 0 and ii < nx) and (jj >=0 and jj < ny):\n",
    "                    x_neighb = np.array([X[ii, jj], Y[ii, jj]])\n",
    "                    stretch = np.linalg.norm(x_center - x_neighb) - l\n",
    "                    acc = - k * l0 / l * stretch * (x_center - x_neighb) / np.linalg.norm(x_center - x_neighb)\n",
    "                    acc_X[i, j] += acc[0]\n",
    "                    acc_Y[i, j] += acc[1]\n",
    "    return grid2linear(acc_X, acc_Y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "from functools import partial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc_func_grid = partial(acc_func_grid_template, nx=nx, ny=ny, k=20.)\n",
    "acc_func_grid(linear_repr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "linear_repr[1] += 0.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.00992562, -2.66129819,  0.65137257,  2.        ,  0.        ,\n",
       "        0.        ,  0.0992562 ,  0.62449111, -0.7237473 ,  0.        ,\n",
       "        0.        ,  0.        ])"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc_func_grid(linear_repr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_grid(x, nx, ny, ax=None):\n",
    "    X, Y = linear2grid(x, nx, ny)\n",
    "    if ax is None:\n",
    "        ax = plt.gca()\n",
    "    ax.scatter(X.ravel(), Y.ravel())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ab6iqr1fVF4CD3f6mUldV3VpVX+tW9zH4vNlJ6zNeyzkP+FBV/XdVPQh8CNg+g5ouAq4f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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1178b0908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_grid(linear_repr, nx=nx, ny=ny)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's try this with some random noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx, ny = 3, 3\n",
    "X, Y = build_grid(nx, ny)\n",
    "x0 = grid2linear(X, Y)\n",
    "x0[1] += 0.3\n",
    "xd0 = np.zeros_like(x0)\n",
    "acc_func_grid = partial(acc_func_grid_template, nx=nx, ny=ny, k=10.)\n",
    "grid = VerletSolver(0.01, x0, xd0, acc_func_grid)\n",
    "ts, vals = grid.step_until(2.5, snapshot_dt=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.12652114, -4.82279424,  0.12652114,  0.78487598,  3.        ,\n",
       "        0.78487598,  0.        ,  0.        ,  0.        ,  0.42173715,\n",
       "        0.        , -0.42173715, -1.12125139,  0.        ,  1.12125139,\n",
       "        0.        ,  0.        ,  0.        ])"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "acc_func_grid(x0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's now plot the grid of particles, between its initial and final state in the simulation. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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eEXdExPGIOBERl3zxdkS8LiK+3V7/bESsH3+V1etjnO6NiPmIeKH987Eq6qxSRDwWEb+I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\nyHCXpAL9P9RRuB58sM8iAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11a9f8c18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_grid(vals[0], nx, ny)\n",
    "plot_grid(vals[-1], nx, ny)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    document.getElementById(this.slider_id).max = this.frames.length - 1;\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<div class=\"animation\" align=\"center\">\n",
       "    <img id=\"_anim_img0769fac9e78a443e96223eb8dc96717f\">\n",
       "    <br>\n",
       "    <input id=\"_anim_slider0769fac9e78a443e96223eb8dc96717f\" type=\"range\" style=\"width:350px\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           onchange=\"anim0769fac9e78a443e96223eb8dc96717f.set_frame(parseInt(this.value));\"></input>\n",
       "    <br>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.previous_frame()\">\n",
       "        <i class=\"fa fa-step-backward\"></i></button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.reverse_animation()\">\n",
       "        <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "        </button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim0769fac9e78a443e96223eb8dc96717f.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "  <form action=\"#n\" name=\"_anim_loop_select0769fac9e78a443e96223eb8dc96717f\" class=\"anim_control\">\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"once\" > Once </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"loop\" checked> Loop </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"reflect\" > Reflect </input>\n",
       "  </form>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img0769fac9e78a443e96223eb8dc96717f\";\n",
       "    var slider_id = \"_anim_slider0769fac9e78a443e96223eb8dc96717f\";\n",
       "    var loop_select_id = \"_anim_loop_select0769fac9e78a443e96223eb8dc96717f\";\n",
       "    var frames = new Array(0);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWgAAAFoCAYAAAB65WHVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
       "AAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMS4xLCBo\\\n",
       "dHRwOi8vbWF0cGxvdGxpYi5vcmcvAOZPmwAACiVJREFUeJzt3T9o3PUfx/F3eqb0kmriJKYmLQhG\\\n",
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       "CCXQAKEEGiCUQAOEEmiAUAINEEqgAUIJNEAogQYIJdAAoQQaIJRAA4QSaIBQAg0QSqABQgk0QCiB\\\n",
       "Bggl0AChBBog1D3DHsB+WFnr1tLqRm1u9Wqm066F+bk6duTgsIcF8Jcm+v1+f9iDuJtW1rq1uLxe\\\n",
       "ve1rN7a1J1v15jOPiTQQrfFTHEurGwNxrqrqbV+rpdWNIY0I4NY0fopjc6u3p+00jykuRlXjj6Bn\\\n",
       "Ou09badZrk9xdbd61a+q7lavFpfXa2WtO+yhwU01PtAL83PVnmwNbGtPtmphfm5II2I/meJilDV+\\\n",
       "iuP6qaxT3PFkiotR1vhAV/0ZaUEeTzOddnV3iLEpLkZB46c4GG+muBhlY3EEzfgyxcUoa/xCFYBR\\\n",
       "ZYoDIJRAA4QSaIBQLhICjdG0Zf0uEgKN0MTfXGmKA2iEJi7rF2igEZq4rF+ggUZo4m+uFGigEZq4\\\n",
       "rN9dHEAjNHFZv7s4AEKZ4gAIJdAAoQQaIJRAA4QSaIBQAg0QSqABQgk0QCiBBggl0AChBBoglEAD\\\n",
       "hBJogFACDRBKoAFCCTRAKIEGCCXQAKEEGiCUQAOEEmiAUAINEEqgAUIJNEAogQYIJdAAoQQaIJRA\\\n",
       "A4QSaIBQAg0QSqABQgk0QCiBBggl0AChBBoglEADhBJogFACDRBKoAFCCTRAKIEGCCXQAKEEGiCU\\\n",
       "QAOEEmiAUAINEEqgAUIJNEAogQYIJdAAoQQaIJRAA4QSaIBQAg0Q6n+5dY6JiobkDAAAAABJRU5E\\\n",
       "rkJggg==\\\n",
       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim0769fac9e78a443e96223eb8dc96717f = new Animation(frames, img_id, slider_id, 100.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_grid(vals[index], nx, ny, ax=ax)\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "    ax.axis('off')\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What if we give some of the particles a specific directional nudge? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx, ny = 11, 12\n",
    "X, Y = build_grid(nx, ny)\n",
    "x0 = grid2linear(X, Y)\n",
    "x0[:ny] += 0.6\n",
    "x0[nx*ny:nx*ny+ny] += 0.2\n",
    "xd0 = np.zeros_like(x0)\n",
    "acc_func_grid = partial(acc_func_grid_template, nx=nx, ny=ny, k=1.)\n",
    "grid = VerletSolver(0.05, x0, xd0, acc_func_grid)\n",
    "ts, vals = grid.step_until(20, snapshot_dt=0.05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EjonIsdHR0dJH6kdI3sYMuChSUzEkIJsu5augDRP7Zhx8cZnAhol9obx/EC4CEUbGEuif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\nA+oppgxJcWhFAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11ad29908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_grid(vals[0], nx, ny)\n",
    "plot_grid(vals[-1], nx, ny)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    document.getElementById(this.slider_id).max = this.frames.length - 1;\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<div class=\"animation\" align=\"center\">\n",
       "    <img id=\"_anim_img5cffc7a57e9d477c8493b78809a973ea\">\n",
       "    <br>\n",
       "    <input id=\"_anim_slider5cffc7a57e9d477c8493b78809a973ea\" type=\"range\" style=\"width:350px\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           onchange=\"anim5cffc7a57e9d477c8493b78809a973ea.set_frame(parseInt(this.value));\"></input>\n",
       "    <br>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.previous_frame()\">\n",
       "        <i class=\"fa fa-step-backward\"></i></button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.reverse_animation()\">\n",
       "        <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "        </button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim5cffc7a57e9d477c8493b78809a973ea.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "  <form action=\"#n\" name=\"_anim_loop_select5cffc7a57e9d477c8493b78809a973ea\" class=\"anim_control\">\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"once\" > Once </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"loop\" checked> Loop </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"reflect\" > Reflect </input>\n",
       "  </form>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img5cffc7a57e9d477c8493b78809a973ea\";\n",
       "    var slider_id = \"_anim_slider5cffc7a57e9d477c8493b78809a973ea\";\n",
       "    var loop_select_id = \"_anim_loop_select5cffc7a57e9d477c8493b78809a973ea\";\n",
       "    var frames = new Array(0);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWgAAAFoCAYAAAB65WHVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
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       "kJSCBkjq/wAI/ONxTGadFQAAAABJRU5ErkJggg==\\\n",
       "\"\n",
       "  frames[1] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWgAAAFoCAYAAAB65WHVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
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       "UP8DOEhcuUc5Fp8AAAAASUVORK5CYII=\\\n",
       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim5cffc7a57e9d477c8493b78809a973ea = new Animation(frames, img_id, slider_id, 100.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 40\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_grid(vals[index], nx, ny, ax=ax)\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "    ax.axis([-1, nx+1, -1, ny+1])\n",
    "    ax.axis('off')\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another version might be to add random movements to the particles."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "nx, ny = 11, 12\n",
    "X, Y = build_grid(nx, ny)\n",
    "x0 = grid2linear(X, Y)\n",
    "x0 += np.random.randn(*x0.shape) / 10\n",
    "xd0 = np.zeros_like(x0)\n",
    "acc_func_grid = partial(acc_func_grid_template, nx=nx, ny=ny, k=1.)\n",
    "grid = VerletSolver(0.05, x0, xd0, acc_func_grid)\n",
    "ts, vals = grid.step_until(20, snapshot_dt=0.05)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<link rel=\"stylesheet\"\n",
       "href=\"https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/\n",
       "css/font-awesome.min.css\">\n",
       "<script language=\"javascript\">\n",
       "  /* Define the Animation class */\n",
       "  function Animation(frames, img_id, slider_id, interval, loop_select_id){\n",
       "    this.img_id = img_id;\n",
       "    this.slider_id = slider_id;\n",
       "    this.loop_select_id = loop_select_id;\n",
       "    this.interval = interval;\n",
       "    this.current_frame = 0;\n",
       "    this.direction = 0;\n",
       "    this.timer = null;\n",
       "    this.frames = new Array(frames.length);\n",
       "\n",
       "    for (var i=0; i<frames.length; i++)\n",
       "    {\n",
       "     this.frames[i] = new Image();\n",
       "     this.frames[i].src = frames[i];\n",
       "    }\n",
       "    document.getElementById(this.slider_id).max = this.frames.length - 1;\n",
       "    this.set_frame(this.current_frame);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.get_loop_state = function(){\n",
       "    var button_group = document[this.loop_select_id].state;\n",
       "    for (var i = 0; i < button_group.length; i++) {\n",
       "        var button = button_group[i];\n",
       "        if (button.checked) {\n",
       "            return button.value;\n",
       "        }\n",
       "    }\n",
       "    return undefined;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.set_frame = function(frame){\n",
       "    this.current_frame = frame;\n",
       "    document.getElementById(this.img_id).src =\n",
       "            this.frames[this.current_frame].src;\n",
       "    document.getElementById(this.slider_id).value = this.current_frame;\n",
       "  }\n",
       "\n",
       "  Animation.prototype.next_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.previous_frame = function()\n",
       "  {\n",
       "    this.set_frame(Math.max(0, this.current_frame - 1));\n",
       "  }\n",
       "\n",
       "  Animation.prototype.first_frame = function()\n",
       "  {\n",
       "    this.set_frame(0);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.last_frame = function()\n",
       "  {\n",
       "    this.set_frame(this.frames.length - 1);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.slower = function()\n",
       "  {\n",
       "    this.interval /= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.faster = function()\n",
       "  {\n",
       "    this.interval *= 0.7;\n",
       "    if(this.direction > 0){this.play_animation();}\n",
       "    else if(this.direction < 0){this.reverse_animation();}\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_forward = function()\n",
       "  {\n",
       "    this.current_frame += 1;\n",
       "    if(this.current_frame < this.frames.length){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.first_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.last_frame();\n",
       "        this.reverse_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.last_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.anim_step_reverse = function()\n",
       "  {\n",
       "    this.current_frame -= 1;\n",
       "    if(this.current_frame >= 0){\n",
       "      this.set_frame(this.current_frame);\n",
       "    }else{\n",
       "      var loop_state = this.get_loop_state();\n",
       "      if(loop_state == \"loop\"){\n",
       "        this.last_frame();\n",
       "      }else if(loop_state == \"reflect\"){\n",
       "        this.first_frame();\n",
       "        this.play_animation();\n",
       "      }else{\n",
       "        this.pause_animation();\n",
       "        this.first_frame();\n",
       "      }\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.pause_animation = function()\n",
       "  {\n",
       "    this.direction = 0;\n",
       "    if (this.timer){\n",
       "      clearInterval(this.timer);\n",
       "      this.timer = null;\n",
       "    }\n",
       "  }\n",
       "\n",
       "  Animation.prototype.play_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = 1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_forward();\n",
       "    }, this.interval);\n",
       "  }\n",
       "\n",
       "  Animation.prototype.reverse_animation = function()\n",
       "  {\n",
       "    this.pause_animation();\n",
       "    this.direction = -1;\n",
       "    var t = this;\n",
       "    if (!this.timer) this.timer = setInterval(function() {\n",
       "        t.anim_step_reverse();\n",
       "    }, this.interval);\n",
       "  }\n",
       "</script>\n",
       "\n",
       "<div class=\"animation\" align=\"center\">\n",
       "    <img id=\"_anim_img432cfd117bd84c17b870b9a2b65bbd74\">\n",
       "    <br>\n",
       "    <input id=\"_anim_slider432cfd117bd84c17b870b9a2b65bbd74\" type=\"range\" style=\"width:350px\"\n",
       "           name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\"\n",
       "           onchange=\"anim432cfd117bd84c17b870b9a2b65bbd74.set_frame(parseInt(this.value));\"></input>\n",
       "    <br>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.slower()\"><i class=\"fa fa-minus\"></i></button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.first_frame()\"><i class=\"fa fa-fast-backward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.previous_frame()\">\n",
       "        <i class=\"fa fa-step-backward\"></i></button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.reverse_animation()\">\n",
       "        <i class=\"fa fa-play fa-flip-horizontal\"></i></button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.pause_animation()\"><i class=\"fa fa-pause\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.play_animation()\"><i class=\"fa fa-play\"></i>\n",
       "        </button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.next_frame()\"><i class=\"fa fa-step-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.last_frame()\"><i class=\"fa fa-fast-forward\">\n",
       "        </i></button>\n",
       "    <button onclick=\"anim432cfd117bd84c17b870b9a2b65bbd74.faster()\"><i class=\"fa fa-plus\"></i></button>\n",
       "  <form action=\"#n\" name=\"_anim_loop_select432cfd117bd84c17b870b9a2b65bbd74\" class=\"anim_control\">\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"once\" > Once </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"loop\" checked> Loop </input>\n",
       "    <input type=\"radio\" name=\"state\"\n",
       "           value=\"reflect\" > Reflect </input>\n",
       "  </form>\n",
       "</div>\n",
       "\n",
       "\n",
       "<script language=\"javascript\">\n",
       "  /* Instantiate the Animation class. */\n",
       "  /* The IDs given should match those used in the template above. */\n",
       "  (function() {\n",
       "    var img_id = \"_anim_img432cfd117bd84c17b870b9a2b65bbd74\";\n",
       "    var slider_id = \"_anim_slider432cfd117bd84c17b870b9a2b65bbd74\";\n",
       "    var loop_select_id = \"_anim_loop_select432cfd117bd84c17b870b9a2b65bbd74\";\n",
       "    var frames = new Array(0);\n",
       "    \n",
       "  frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWgAAAFoCAYAAAB65WHVAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n",
       "AAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMS4xLCBo\\\n",
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       "KMcw7zIihh8UhCkG/cGiC6Ggi+qmmG5koBNnyKsk6DpEb06I4ekiCCnCi0LUFA/khYGcORCRmk7H\\\n",
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       "CGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgA\\\n",
       "CIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqA\\\n",
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       "CGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACIqABoCgCGgA\\\n",
       "CIqABoCgCGgACIqABoCgCGgACIqABoCgCGgACOr/GJODlGj0VRsAAAAASUVORK5CYII=\\\n",
       "\"\n",
       "\n",
       "\n",
       "    /* set a timeout to make sure all the above elements are created before\n",
       "       the object is initialized. */\n",
       "    setTimeout(function() {\n",
       "        anim432cfd117bd84c17b870b9a2b65bbd74 = new Animation(frames, img_id, slider_id, 100.0,\n",
       "                                 loop_select_id);\n",
       "    }, 0);\n",
       "  })()\n",
       "</script>\n"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 40\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    plot_grid(vals[index], nx, ny, ax=ax)\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "    ax.axis([-1, nx+1, -1, ny+1])\n",
    "    ax.axis('off')\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)\n",
    "HTML(animation.to_jshtml())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lennard Jones particles "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Trying to mess with the png encoding in to_jshtml "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulating the spring."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 320,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.array([0, -1], dtype=np.float)\n",
    "xd0 = np.array([0.0, 0.1], dtype=np.float)\n",
    "spring = VerletSolver(0.01, x0, xd0, acc_func_spring)\n",
    "ts, vals = spring.step_until(12, snapshot_dt=0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Making the animation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 321,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(figsize=(5, 5))\n",
    "\n",
    "FRAMES = 50\n",
    "def animate(i):\n",
    "    index = np.argmin((np.linspace(0, 1, vals.shape[0]) - i / FRAMES)**2)\n",
    "    ax.clear()\n",
    "    ax.plot(vals[index, 0], vals[index, 1], 'o')\n",
    "    ax.plot(vals[index-5:index+1, 0], vals[index-5:index+1, 1], '-')\n",
    "    ax.set_xlim(-1, 1)\n",
    "    ax.set_ylim(-1, 1)\n",
    "    ax.set_title(f\"t = {ts[index]:.2f}\")\n",
    "\n",
    "animation = mpl_anim.FuncAnimation(fig, animate, init_func=lambda: None,\n",
    "                        frames=FRAMES, interval=100, blit=False)\n",
    "plt.close(fig)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 322,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 322,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plt.rcParams['animation.html_args']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 323,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'h264'"
      ]
     },
     "execution_count": 323,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plt.rcParams['animation.codec']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 324,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "execution_count": 324,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "HTML(animation.to_html5_video())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 325,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.animation.HTMLWriter at 0x119209390>"
      ]
     },
     "execution_count": 325,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "mpl_anim.HTMLWriter(fps=10, extra_args={'frame_format':'jpeg'})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 326,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpl_anim.Animation.to_jshtml = mpl_anim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import tempfile\n",
    "HTMLWriter = mpl_anim.HTMLWriter\n",
    "import os\n",
    "\n",
    "def to_jshtml_jpeg(self, fps=None, embed_frames=True, default_mode=None):\n",
    "    \"\"\"Generate HTML representation of the animation\"\"\"\n",
    "    if fps is None and hasattr(self, '_interval'):\n",
    "        # Convert interval in ms to frames per second\n",
    "        fps = 1000 / self._interval\n",
    "\n",
    "    # If we're not given a default mode, choose one base on the value of\n",
    "    # the repeat attribute\n",
    "    if default_mode is None:\n",
    "        default_mode = 'loop' if self.repeat else 'once'\n",
    "\n",
    "    if hasattr(self, \"_html_representation\"):\n",
    "        return self._html_representation\n",
    "    else:\n",
    "        # Can't open a second time while opened on windows. So we avoid\n",
    "        # deleting when closed, and delete manually later.\n",
    "        with tempfile.NamedTemporaryFile(suffix='.html',\n",
    "                                         delete=False) as f:\n",
    "            self.save(f.name, writer=HTMLWriter(fps=fps,\n",
    "                                                embed_frames=embed_frames,\n",
    "                                                default_mode=default_mode,\n",
    "                                                extra_args={'frame_format':'jpeg'}))\n",
    "        # Re-open and get content\n",
    "        with open(f.name) as fobj:\n",
    "            html = fobj.read()\n",
    "\n",
    "        # Now we can delete\n",
    "        os.remove(f.name)\n",
    "\n",
    "        self._html_representation = html\n",
    "    return html"
   ]
  }
 ],
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