Periodic Trajectories and Topology of the Integrable Boltzmann System

TL;DR

This paper analyzes the integrable Boltzmann billiard system with a linear boundary under gravity, deriving conditions for periodic trajectories, describing caustics, and exploring phase space topology using Fomenko graphs.

Contribution

It introduces analytic Cayley-type conditions for periodicity and provides a topological analysis of the phase space for this specific billiard system.

Findings

Derived explicit conditions for periodic trajectories.

Provided geometric descriptions of caustics.

Analyzed phase space topology with Fomenko graphs.

Abstract

We consider the Boltzmann system corresponding to the motion of a billiard with a linear boundary under the influence of a gravitational field. We derive analytic conditions of Cayley's type for periodicity of its trajectories and provide geometric descriptions of caustics. The topology of the phase space is discussed using Fomenko graphs.