Billiard table
Poincaré section (θ, p=cosφ)
Shape
Circle: completely integrable — all orbits are KAM tori (horizontal lines in Poincaré section).
Ellipse: integrable — two families of caustics (ellipses + hyperbolas).
Stadium: chaotic (Bunimovich 1979) — ergodic, positive Lyapunov exponent.
Ellipse: integrable — two families of caustics (ellipses + hyperbolas).
Stadium: chaotic (Bunimovich 1979) — ergodic, positive Lyapunov exponent.