Integrability of Billiards Inside Cones as a Discrete-Time Hamiltonian System

TL;DR

This paper demonstrates that billiards inside convex cones over smooth manifolds are completely integrable as discrete-time Hamiltonian systems, expanding the class of known integrable billiard systems.

Contribution

It proves the complete integrability of billiards inside convex cones over smooth manifolds, showing they admit sufficient first integrals in involution.

Findings

Billiards inside convex cones are superintegrable.

The system admits n-1 independent first integrals in involution.

The billiard system is an example of integrability beyond quadrics.

Abstract

In this paper, we continue to study billiards inside cones $K\subset \mathbb{R}^n$ over strictly convex closed $C^3$ manifolds with non-degenerate second fundamental form. Recently we proved that the billiard is superintegrable, i.e., the billiard admits first integrals whose values uniquely determine all billiard trajectories. In this paper we prove that this billiard system admits $n-1$ independent first integrals in involution. Consequently, the system is completely integrable as a discrete-time Hamiltonian system. This provides an example of an integrable billiard where the billiard table is neither a quadric nor consists of pieces of quadrics.