Rectangle Billiard (Integrable)

A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \(\mathbf{v}' = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}\), whenever it strikes the boundary. Here the boundary is the rectangle \(|x| \le w/2,\ |y| \le h/2\) of width \(w\) and height \(h\), whose walls are all axis-aligned, so \(\mathbf{n}\) is always \(\pm\hat{x}\) or \(\pm\hat{y}\) – reflections off the horizontal and vertical walls therefore only ever flip the sign of one velocity component. The rectangular billiard is integrable: each individual ray only ever visits a handful of discrete \(\sin\phi\) values (the sine of the angle between the outgoing velocity and the local boundary tangent). This example plots a single trajectory and a Poincare section built from several rays launched at different angles.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.chaos.systems.billiards import RectangleBilliard
from physicskit.chaos.visualizers.dynamic_plots import animate_billiard_trajectory
from physicskit.chaos.visualizers.phase_space import plot_billiard_trajectory, plot_poincare_section

billiard = RectangleBilliard(width=2.0, height=1.0)

Animation

anim = animate_billiard_trajectory(billiard, pos=billiard.sample_interior_point(), vel=(1.0, 0.35), 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("rectangle_billiard_animation.gif", writer="pillow", fps=25)

Trajectory

fig, ax = plot_billiard_trajectory(billiard, pos=billiard.sample_interior_point(), vel=np.array([1.0, 0.35]), n_bounces=60)

Poincare section

Each ray only ever hits walls at two possible incidence angles (one for the horizontal walls, one for the vertical walls), so the section is made up of a discrete set of horizontal line segments rather than the dense chaotic sea seen in defocusing billiards.

fig, ax = plot_poincare_section(billiard, n_rays=25, n_bounces=150)

plt.show()