Ergodic properties of a particle moving elastically inside a polygon

Abstract

The flow of a classical particle bouncing elastically inside an arbitrary polygon is investigated. If every interior angle is a rational multiple of π, there exists precisely one isolating integral in addition to the energy; this integral is described in detail; any possible third integral is nonisolating. If one or more interior angles is an irrational multiple of π, the second integral becomes everywhere nonisolating and non‐Lebesgue‐measurable, i.e., the second integral disappears. The flow of two hard points bouncing elastically in a finite one‐dimensional box is equivalent to the flow of a point particle moving elastically inside a right triangle having interior angle tan⁻¹ (m₂/m₁)^1/2, so the preceding remarks apply to this model. Nonrigorous arguments are given in support of the notion that the polygon model is ergodic and mixing, but is not a C‐system.