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

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.