On Elliptical Billiards in the Lobachevsky Space and associated Geodesic Hierarchies

TL;DR

This paper extends the theory of elliptical billiards to Lobachevsky space, deriving conditions for periodic trajectories and revealing a hierarchy linking Euclidean and hyperbolic billiards through integrability.

Contribution

It derives Cayley's conditions for hyperbolic billiards and explains their similarity to Euclidean cases using geodesically equivalent metrics, establishing a hierarchy of integrable billiards.

Findings

Cayley's conditions for Lobachevsky billiards are similar to Euclidean cases.

Lobachevsky and Euclidean billiards are part of a hierarchy of integrable systems.

Theoretical connection via geodesically equivalent metrics explains the observed coincidences.

Abstract

We derive Cayley's type conditions for periodical trajectories for the billiard within an ellipsoid in the Lobachevsky space. It appears that these new conditions are of the same form as those obtained before for the Euclidean case. We explain this coincidence by using theory of geodesically equivalent metrics and show that Lobachevsky and Euclidean elliptic billiards can be naturally considered as a part of a hierarchy of integrable elliptical billiards.