Circular, elliptic and oval billiards in a gravitational field

TL;DR

This paper investigates the classical dynamics of particles in gravitational billiards with circular, elliptic, and oval boundaries, revealing ergodic behavior at certain energies and analyzing stability, bifurcations, and velocity growth under boundary variations.

Contribution

It provides a detailed analytical and numerical study of gravitational billiards, including stability analysis, bifurcation points, and velocity growth, extending understanding of chaotic and ergodic regimes.

Findings

Ergodic behavior at specific energies in all models.

Exact bifurcation points for stability changes.

Unlimited velocity growth in oval billiards with boundary dynamics.

Abstract

We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the energy regimes is made. The linear stability of fixed points is studied, yielding exact analytical expressions for parameter values at which a period-doubling bifurcation occurs. The dynamics is apparently ergodic at certain energies in all three models, in contrast to the regularity of the circular and elliptic billiard dynamics in the field-free case. This finding is confirmed using a sensitive test involving Lyapunov weighted dynamics. In the last part of the paper a time dependence is introduced in the billiard boundary, where it is shown that for the circular billiard the average velocity saturates for zero gravitational force but in the presence of gravitational it increases with a very slow growth rate, which may be explained using Arnold diffusion. For the oval billiard, where chaos is present in the static case, the particle has an unlimited velocity growth with an exponent of approximately 1/6.