text: #	
  #	References
  #	1. url https://matplotlib.org/gallery/animation
  #	   /double_pendulum_animated_sgskip.html [20210205].
  #	
  
  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()