Pseudo-Euclidean Billiards within Confocal Curves on the Hyperboloid of One Sheet

TL;DR

This paper studies billiard dynamics within confocal conic boundaries on a hyperboloid in Minkowski space, demonstrating integrability, generalizing classical theorems, and deriving conditions for periodic trajectories.

Contribution

It introduces two types of confocal families on a hyperboloid and proves integrability of billiards within these, extending classical geometric results to a Minkowski setting.

Findings

Billiard systems are integrable within confocal conics on hyperboloids.

A generalization of the Poncelet theorem is established.

Cayley-type conditions for periodic trajectories are derived.

Abstract

We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We show that there are two types of confocal families in such setting. Using an algebro-geometric integration technique, we prove that the billiard within generalized ellipses of each type is integrable in the sense of Liouville. Further, we prove a generalization of the Poncelet theorem and derive Cayley-type conditions for periodic trajectories and explore geometric consequences.