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.