
.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "gallery/animation/double_pendulum.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. meta::
        :keywords: codex

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_gallery_animation_double_pendulum.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_gallery_animation_double_pendulum.py:


===========================
The double pendulum problem
===========================

This animation illustrates the double pendulum problem.

Double pendulum formula translated from the C code at
http://www.physics.usyd.edu.au/~wheat/dpend_html/solve_dpend.c

Output generated via `matplotlib.animation.Animation.to_jshtml`.

.. GENERATED FROM PYTHON SOURCE LINES 13-111



.. container:: sphx-glr-animation

    .. raw:: html

        
     <link rel="stylesheet"
     href="https://maxcdn.bootstrapcdn.com/font-awesome/4.4.0/css/font-awesome.min.css">
     <script language="javascript">
       function isInternetExplorer() {
         ua = navigator.userAgent;
         /* MSIE used to detect old browsers and Trident used to newer ones*/
         return ua.indexOf("MSIE ") > -1 || ua.indexOf("Trident/") > -1;
       }

       /* Define the Animation class */
       function Animation(frames, img_id, slider_id, interval, loop_select_id){
         this.img_id = img_id;
         this.slider_id = slider_id;
         this.loop_select_id = loop_select_id;
         this.interval = interval;
         this.current_frame = 0;
         this.direction = 0;
         this.timer = null;
         this.frames = new Array(frames.length);

         for (var i=0; i<frames.length; i++)
         {
          this.frames[i] = new Image();
          this.frames[i].src = frames[i];
         }
         var slider = document.getElementById(this.slider_id);
         slider.max = this.frames.length - 1;
         if (isInternetExplorer()) {
             // switch from oninput to onchange because IE <= 11 does not conform
             // with W3C specification. It ignores oninput and onchange behaves
             // like oninput. In contrast, Microsoft Edge behaves correctly.
             slider.setAttribute('onchange', slider.getAttribute('oninput'));
             slider.setAttribute('oninput', null);
         }
         this.set_frame(this.current_frame);
       }

       Animation.prototype.get_loop_state = function(){
         var button_group = document[this.loop_select_id].state;
         for (var i = 0; i < button_group.length; i++) {
             var button = button_group[i];
             if (button.checked) {
                 return button.value;
             }
         }
         return undefined;
       }

       Animation.prototype.set_frame = function(frame){
         this.current_frame = frame;
         document.getElementById(this.img_id).src =
                 this.frames[this.current_frame].src;
         document.getElementById(this.slider_id).value = this.current_frame;
       }

       Animation.prototype.next_frame = function()
       {
         this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));
       }

       Animation.prototype.previous_frame = function()
       {
         this.set_frame(Math.max(0, this.current_frame - 1));
       }

       Animation.prototype.first_frame = function()
       {
         this.set_frame(0);
       }

       Animation.prototype.last_frame = function()
       {
         this.set_frame(this.frames.length - 1);
       }

       Animation.prototype.slower = function()
       {
         this.interval /= 0.7;
         if(this.direction > 0){this.play_animation();}
         else if(this.direction < 0){this.reverse_animation();}
       }

       Animation.prototype.faster = function()
       {
         this.interval *= 0.7;
         if(this.direction > 0){this.play_animation();}
         else if(this.direction < 0){this.reverse_animation();}
       }

       Animation.prototype.anim_step_forward = function()
       {
         this.current_frame += 1;
         if(this.current_frame < this.frames.length){
           this.set_frame(this.current_frame);
         }else{
           var loop_state = this.get_loop_state();
           if(loop_state == "loop"){
             this.first_frame();
           }else if(loop_state == "reflect"){
             this.last_frame();
             this.reverse_animation();
           }else{
             this.pause_animation();
             this.last_frame();
           }
         }
       }

       Animation.prototype.anim_step_reverse = function()
       {
         this.current_frame -= 1;
         if(this.current_frame >= 0){
           this.set_frame(this.current_frame);
         }else{
           var loop_state = this.get_loop_state();
           if(loop_state == "loop"){
             this.last_frame();
           }else if(loop_state == "reflect"){
             this.first_frame();
             this.play_animation();
           }else{
             this.pause_animation();
             this.first_frame();
           }
         }
       }

       Animation.prototype.pause_animation = function()
       {
         this.direction = 0;
         if (this.timer){
           clearInterval(this.timer);
           this.timer = null;
         }
       }

       Animation.prototype.play_animation = function()
       {
         this.pause_animation();
         this.direction = 1;
         var t = this;
         if (!this.timer) this.timer = setInterval(function() {
             t.anim_step_forward();
         }, this.interval);
       }

       Animation.prototype.reverse_animation = function()
       {
         this.pause_animation();
         this.direction = -1;
         var t = this;
         if (!this.timer) this.timer = setInterval(function() {
             t.anim_step_reverse();
         }, this.interval);
       }
     </script>

     <style>
     .animation {
         display: inline-block;
         text-align: center;
     }
     input[type=range].anim-slider {
         width: 374px;
         margin-left: auto;
         margin-right: auto;
     }
     .anim-buttons {
         margin: 8px 0px;
     }
     .anim-buttons button {
         padding: 0;
         width: 36px;
     }
     .anim-state label {
         margin-right: 8px;
     }
     .anim-state input {
         margin: 0;
         vertical-align: middle;
     }
     </style>

     <div class="animation">
       <img id="_anim_img91c12421621a4cd7a3743535ebaf7d4f">
       <div class="anim-controls">
         <input id="_anim_slider91c12421621a4cd7a3743535ebaf7d4f" type="range" class="anim-slider"
                name="points" min="0" max="1" step="1" value="0"
                oninput="anim91c12421621a4cd7a3743535ebaf7d4f.set_frame(parseInt(this.value));">
         <div class="anim-buttons">
           <button title="Decrease speed" aria-label="Decrease speed" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.slower()">
               <i class="fa fa-minus"></i></button>
           <button title="First frame" aria-label="First frame" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.first_frame()">
             <i class="fa fa-fast-backward"></i></button>
           <button title="Previous frame" aria-label="Previous frame" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.previous_frame()">
               <i class="fa fa-step-backward"></i></button>
           <button title="Play backwards" aria-label="Play backwards" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.reverse_animation()">
               <i class="fa fa-play fa-flip-horizontal"></i></button>
           <button title="Pause" aria-label="Pause" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.pause_animation()">
               <i class="fa fa-pause"></i></button>
           <button title="Play" aria-label="Play" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.play_animation()">
               <i class="fa fa-play"></i></button>
           <button title="Next frame" aria-label="Next frame" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.next_frame()">
               <i class="fa fa-step-forward"></i></button>
           <button title="Last frame" aria-label="Last frame" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.last_frame()">
               <i class="fa fa-fast-forward"></i></button>
           <button title="Increase speed" aria-label="Increase speed" onclick="anim91c12421621a4cd7a3743535ebaf7d4f.faster()">
               <i class="fa fa-plus"></i></button>
         </div>
         <form title="Repetition mode" aria-label="Repetition mode" action="#n" name="_anim_loop_select91c12421621a4cd7a3743535ebaf7d4f"
               class="anim-state">
           <input type="radio" name="state" value="once" id="_anim_radio1_91c12421621a4cd7a3743535ebaf7d4f"
                  >
           <label for="_anim_radio1_91c12421621a4cd7a3743535ebaf7d4f">Once</label>
           <input type="radio" name="state" value="loop" id="_anim_radio2_91c12421621a4cd7a3743535ebaf7d4f"
                  checked>
           <label for="_anim_radio2_91c12421621a4cd7a3743535ebaf7d4f">Loop</label>
           <input type="radio" name="state" value="reflect" id="_anim_radio3_91c12421621a4cd7a3743535ebaf7d4f"
                  >
           <label for="_anim_radio3_91c12421621a4cd7a3743535ebaf7d4f">Reflect</label>
         </form>
       </div>
     </div>


     <script language="javascript">
       /* Instantiate the Animation class. */
       /* The IDs given should match those used in the template above. */
       (function() {
         var img_id = "_anim_img91c12421621a4cd7a3743535ebaf7d4f";
         var slider_id = "_anim_slider91c12421621a4cd7a3743535ebaf7d4f";
         var loop_select_id = "_anim_loop_select91c12421621a4cd7a3743535ebaf7d4f";
         var frames = new Array(250);
    
       frames[0] = "data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfQAAAGQCAYAAABYs5LGAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90\
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     "


         /* set a timeout to make sure all the above elements are created before
            the object is initialized. */
         setTimeout(function() {
             anim91c12421621a4cd7a3743535ebaf7d4f = new Animation(frames, img_id, slider_id, 10.0,
                                      loop_select_id);
         }, 0);
       })()
     </script>






.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    from numpy import cos, sin

    import matplotlib.animation as animation

    G = 9.8  # acceleration due to gravity, in m/s^2
    L1 = 1.0  # length of pendulum 1 in m
    L2 = 1.0  # length of pendulum 2 in m
    L = L1 + L2  # maximal length of the combined pendulum
    M1 = 1.0  # mass of pendulum 1 in kg
    M2 = 1.0  # mass of pendulum 2 in kg
    t_stop = 2.5  # how many seconds to simulate
    history_len = 500  # how many trajectory points to display


    def derivs(t, state):
        dydx = np.zeros_like(state)

        dydx[0] = state[1]

        delta = state[2] - state[0]
        den1 = (M1+M2) * L1 - M2 * L1 * cos(delta) * cos(delta)
        dydx[1] = ((M2 * L1 * state[1] * state[1] * sin(delta) * cos(delta)
                    + M2 * G * sin(state[2]) * cos(delta)
                    + M2 * L2 * state[3] * state[3] * sin(delta)
                    - (M1+M2) * G * sin(state[0]))
                   / den1)

        dydx[2] = state[3]

        den2 = (L2/L1) * den1
        dydx[3] = ((- M2 * L2 * state[3] * state[3] * sin(delta) * cos(delta)
                    + (M1+M2) * G * sin(state[0]) * cos(delta)
                    - (M1+M2) * L1 * state[1] * state[1] * sin(delta)
                    - (M1+M2) * G * sin(state[2]))
                   / den2)

        return dydx

    # create a time array from 0..t_stop sampled at 0.02 second steps
    dt = 0.01
    t = np.arange(0, t_stop, dt)

    # th1 and th2 are the initial angles (degrees)
    # w10 and w20 are the initial angular velocities (degrees per second)
    th1 = 120.0
    w1 = 0.0
    th2 = -10.0
    w2 = 0.0

    # initial state
    state = np.radians([th1, w1, th2, w2])

    # integrate the ODE using Euler's method
    y = np.empty((len(t), 4))
    y[0] = state
    for i in range(1, len(t)):
        y[i] = y[i - 1] + derivs(t[i - 1], y[i - 1]) * dt

    # A more accurate estimate could be obtained e.g. using scipy:
    #
    #   y = scipy.integrate.solve_ivp(derivs, t[[0, -1]], state, t_eval=t).y.T

    x1 = L1*sin(y[:, 0])
    y1 = -L1*cos(y[:, 0])

    x2 = L2*sin(y[:, 2]) + x1
    y2 = -L2*cos(y[:, 2]) + y1

    fig = plt.figure(figsize=(5, 4))
    ax = fig.add_subplot(autoscale_on=False, xlim=(-L, L), ylim=(-L, 1.))
    ax.set_aspect('equal')
    ax.grid()

    line, = ax.plot([], [], 'o-', lw=2)
    trace, = ax.plot([], [], '.-', lw=1, ms=2)
    time_template = 'time = %.1fs'
    time_text = ax.text(0.05, 0.9, '', transform=ax.transAxes)


    def animate(i):
        thisx = [0, x1[i], x2[i]]
        thisy = [0, y1[i], y2[i]]

        history_x = x2[:i]
        history_y = y2[:i]

        line.set_data(thisx, thisy)
        trace.set_data(history_x, history_y)
        time_text.set_text(time_template % (i*dt))
        return line, trace, time_text


    ani = animation.FuncAnimation(
        fig, animate, len(y), interval=dt*1000, blit=True)
    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 22.463 seconds)


.. _sphx_glr_download_gallery_animation_double_pendulum.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: double_pendulum.ipynb <double_pendulum.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: double_pendulum.py <double_pendulum.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: double_pendulum.zip <double_pendulum.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_
