Ergodic properties of a particle moving elastically inside a polygon

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.