

.. _sphx_glr_gallery_animation_double_pendulum_animated_sgskip.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



.. code-block:: python


    from numpy import sin, cos
    import numpy as np
    import matplotlib.pyplot as plt
    import scipy.integrate as integrate
    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
    M1 = 1.0  # mass of pendulum 1 in kg
    M2 = 1.0  # mass of pendulum 2 in kg


    def derivs(state, t):

        dydx = np.zeros_like(state)
        dydx[0] = state[1]

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

        dydx[2] = state[3]

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

        return dydx

    # create a time array from 0..100 sampled at 0.05 second steps
    dt = 0.05
    t = np.arange(0.0, 20, 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 your ODE using scipy.integrate.
    y = integrate.odeint(derivs, state, 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()
    ax = fig.add_subplot(111, autoscale_on=False, xlim=(-2, 2), ylim=(-2, 2))
    ax.set_aspect('equal')
    ax.grid()

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


    def init():
        line.set_data([], [])
        time_text.set_text('')
        return line, time_text


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

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

    ani = animation.FuncAnimation(fig, animate, np.arange(1, len(y)),
                                  interval=25, blit=True, init_func=init)

    # ani.save('double_pendulum.mp4', fps=15)
    plt.show()

**Total running time of the script:** ( 0 minutes  0.000 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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

    `Generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
