Keplerian billiards in three dimensions: stability of equilibrium orbits and conditions for chaos

TL;DR

This paper investigates the stability and chaos conditions of equilibrium orbits in three-dimensional Keplerian billiards with different potential models, analyzing how these dynamics relate to topological chaos at high energies.

Contribution

It introduces two models of 3D Keplerian billiards and analyzes the stability of homothetic equilibrium trajectories in relation to chaotic behavior.

Findings

Homothetic trajectories are linked to chaotic dynamics at high energies.

Inner Keplerian potential influences the stability of equilibrium orbits.

Refractive billiards exhibit generalized refraction laws affecting orbit stability.

Abstract

This work presents some results regarding three-dimensional billiards having a non-constant potential of Keplerian type inside a regular domain $D\subset \mathcal R^3$. Two models will be analysed: in the first one, only an inner Keplerian potential is present, and every time the particle encounters the boundary of $D$ is reflected back by keeping constant its tangential component to $\partial D$. The second model is a refractive billiard, where the inner Keplerian potential is coupled with a harmonic outer one; in this case, the interaction with $\partial D$ results in a generalised refraction Snell's law. In both cases, the analysis of a particular type of straight equilibrium trajectories, called \emph{homothetic}, is carried on, and their presence is linked to the topological chaoticity of the dynamics for large inner energies.