Billiard trajectories inside Cones

TL;DR

This paper explores billiard trajectories inside convex cones, proving the existence of cones with trajectories that have infinitely many reflections in finite time, contrasting previous results that all trajectories have finitely many reflections.

Contribution

It introduces $C^2$ convex cones where billiard trajectories can have infinitely many reflections in finite time, expanding understanding of billiard dynamics in convex cones.

Findings

Existence of $C^2$ convex cones with infinitely reflected trajectories.

Estimation of the number of reflections in elliptic cones in $\\mathbb{R}^3$.

Use of two first integrals for reflection count estimation.

Abstract

Recently it was proved that every billiard trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by a $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed $C^3$ submanifold of the hyperplane with nondegenerate second fundamental form. In this paper, we prove the existence of $C^2$ convex cones admitting billiard trajectories with infinitely many reflections in finite time. We also estimate the number of reflections of billiard trajectories in elliptic cones in $\mathbb{R}^3$ using two first integrals.