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arXiv · 2502.01997

Billiard trajectories inside Cones

Andrey E. Mironov·Siyao Yin
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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.

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Andrey E. Mironov, Siyao Yin. 2025-02-04. Billiard trajectories inside Cones. https://arxiv.org/abs/2502.01997

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