Introducing sub-Riemannian and sub-Finsler Billiards

TL;DR

This paper extends billiard dynamics to sub-Finsler geometry, exploring symplectic, variational, and control perspectives, and investigates phenomena like singular boundary points and magnetic billiards in 3D contact spaces.

Contribution

It introduces the concept of billiards in sub-Finsler geometry, providing a unified symplectic and control theoretical framework, and analyzes complex behaviors such as singular reflections and magnetic effects.

Findings

Failure of reflection law at singular boundary points

Existence of gliding and creeping orbits

Periodic trajectories in magnetic billiards

Abstract

We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting, including the failure of the reflection law to be well-defined at singular points of the boundary distribution, the appearance of gliding and creeping orbits, and the behavior of reflections at wavefronts. We then study some concrete tables in 3-dimensional euclidean space endowed with the standard contact structure. These can be interpreted as planar magnetic billiards, of varying magnetic strength, for which the magnetic strength may change under reflection. For each table we provide various results regarding periodic trajectories, gliding orbits, and creeping orbits.