Billiards - Circle, Stadium, Sinai

What you are seeing: a free particle bouncing elastically off the walls of a 2D shape. Three shapes, three very different long-term behaviors. The circle is integrable: the angle of incidence at every wall hit is fixed, so the trajectory traces out a regular caustic and never explores the whole space. The Bunimovich stadium (a rectangle capped with two semicircles) is fully chaotic: any straight segment, however short, is enough to make trajectories diverge exponentially. The Sinai billiard (square with a circular hole at the center) is also chaotic, for the opposite reason: the convex obstacle "defocuses" rays at every reflection.

Specular reflection at each wall: angle of incidence = angle of reflection. The particle has unit speed; the only invariant is the energy (which is constant exactly, machine precision). The trail accumulates over time; click "Reset" to start fresh.

Figure 1. A free particle in three classical billiard geometries (circle, Bunimovich stadium, Sinai). Method: ray-tracing with specular reflection.

geometry

speed6

WHAT TO TRY

Two balls launch from the same point 0.001 rad apart (warm and cool trails). In the stadium and Sinai tables they peel apart almost at once and the bottom plot shoots up exponentially: deterministic chaos.
Switch to the circle or the ellipse: the same twins barely separate and the plot creeps up only linearly. These tables are integrable, so nearby orbits stay nearby.
In the ellipse, launch from a focus and every chord reflects through the other focus, the classic two-focus property.