Acceleration in a nonplanar time-dependent billiard

TL;DR

This paper investigates the dynamical behavior and energy growth of a particle in a sinusoidally shaped, non-planar billiard, analyzing how static and time-dependent geometries influence chaos, regularity, and energy saturation.

Contribution

It introduces a model of a non-planar, time-dependent billiard with sinusoidal geometry and explores how parameters affect particle dynamics and energy growth, revealing conditions for chaos and energy saturation.

Findings

Chaotic static billiard leads to initial energy growth and subsequent saturation.

Regular static billiard results in constant average energy in the time-dependent case.

Particle trajectories can be regular, mixed, or chaotic depending on parameters.

Abstract

We study the dynamical properties of a particle in a non-planar square billiard. The plane of the billiard has a sinusoidal shape. We consider both the static and time-dependent plane. We study the affect of different parameters that control the geometry of the billiard in this model. We consider variations of different parameters of the model and describe how the particle trajectory is affected by these parameters. We also investigate the dynamical behavior of the system in the static condition using its reduced phase plot and show that the dynamics of the particle inside the billiard may be regular, mixed or chaotic. Finally, the problem of the particle energy growth is studied in the billiard with the time-dependent plane. We show that when in the static case, the billiard is chaotic, then the particle energy in the time-dependent billiard grows for small number of collisions, and then it starts to saturate. But when the dynamics of the static case is regular, then the particle average energy in the time-dependent situation stays constant.