Animated Billiard Trajectory

The billiard animated here is the Bunimovich stadium: two straight edges, \(y=\pm r\) for \(|x|\le a\), joined by semicircular caps of radius \(r\) centered at \((\pm a, 0)\). A billiard particle moves in a straight line at constant speed and undergoes specular reflection off this boundary,

\[\mathbf{v}' = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n},\]

whenever it strikes it. physicskit.chaos.visualizers.dynamic_plots.animate_billiard_trajectory() builds a Matplotlib FuncAnimation showing the ball bouncing inside the stadium (left panel) alongside its Poincare section filling in live, point by point, as each new bounce occurs (right panel).

import matplotlib.pyplot as plt
import numpy as np

from physicskit.chaos.systems.billiards import BunimovichStadium
from physicskit.chaos.visualizers.dynamic_plots import animate_billiard_trajectory

billiard = BunimovichStadium(radius=1.0, straight_length=2.0)

Build the animation

Assign the animation to a variable to keep it alive; a reference must survive until it is displayed (plt.show()) or saved (anim.save(...)) or Matplotlib will silently drop it.

anim = animate_billiard_trajectory(billiard, pos=billiard.sample_interior_point(), vel=(0.5, 0.9), n_bounces=100, interval=50)

plt.show()

To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:

anim.save("stadium_animation.gif", writer="pillow", fps=30)

Check

The stadium ray keeps its speed but visits a continuum of incidence angles (chaotic), unlike an integrable billiard.

run = billiard.simulate(billiard.sample_interior_point(), (0.5, 0.9), n_bounces=400)
assert np.ptp(np.hypot(run["vx"], run["vy"])) < 1e-12
assert len(np.unique(np.round(np.abs(run["sin_phi"]), 6))) > 100