Poncelet property and quasi-periodicity of the integrable Boltzmann system

TL;DR

This paper demonstrates that a specific integrable Boltzmann system exhibits Poncelet property and quasi-periodicity, revealing deep geometric structures and stability features in the particle motion under certain forces.

Contribution

It establishes the Poncelet property for the system's orbits and proves the quasi-periodicity conjecture, connecting geometric and dynamical aspects of the integrable Boltzmann system.

Findings

The system's orbits are either all periodic or all non-periodic for given integrals.

The level sets of integrals form elliptic curves, enabling Poncelet property.

Quasi-periodicity allows KAM theory application to the system.

Abstract

We study the motion of a particle in a plane subject to an attractive central force with inverse-square law on one side of a wall at which it is reflected elastically. This model is a special case of a class of systems considered by Boltzmann which was recently shown by Gallavotti and Jauslin to admit a second integral of motion additionally to the energy. By recording the subsequent positions and momenta of the particle as it hits the wall we obtain a three dimensional discrete-time dynamical system. We show that this system has the Poncelet property: if for given generic values of the integrals one orbit is periodic then all orbits for these values are periodic and have the same period. The reason for this is the same as in the case of the Poncelet theorem: the generic level set of the integrals of motion is an elliptic curve, the Poincar\'e map is the composition of two involutions with fixed points and is thus the translation by a fixed element. Another consequence of our result is the proof of a conjecture of Gallavotti and Jauslin on the quasi-periodicity of the integrable Boltzmann system, implying the applicability of KAM perturbation theory to the Boltzmann system with weak centrifugal force.