Integrable Birkhoff Billiards inside Cones

TL;DR

This paper proves that Birkhoff billiards inside convex $C^3$ cones are integrable, providing the first example of such billiards that are neither quadrics nor composed of quadrics, expanding understanding of integrable billiard systems.

Contribution

The paper demonstrates the integrability of Birkhoff billiards inside convex $C^3$ cones, introducing a new class of integrable billiard tables beyond quadrics.

Findings

Billiards inside convex $C^3$ cones admit a quadratic first integral.

Trajectories in such cones have a finite number of reflections.

This is the first known integrable billiard table not based on quadrics.

Abstract

One of the most interesting problems in the theory of Birkhoff billiards is the problem of integrability. In all known examples of integrable billiards, the billiard tables are either conics, quadrics (closed ellipsoids as well as unclosed quadrics like paraboloids or cones), or specific configurations of conics or quadrics. This leads to the natural question: are there other integrable billiards? The Birkhoff conjecture states that if the billiard inside a convex, smooth, closed curve is integrable, then the curve is an ellipse or a circle. In this paper we study the Birkhoff billiard inside a cone in $\mathbb{R}^n$. We prove that the billiard always admits a first integral of degree two in the components of the velocity vector. Using this fact, we prove that every trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by $C^3$ 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. The main result of this paper is the following. We prove that the Birkhoff billiard inside a convex $C^3$ cone is integrable. This is the first example of an integrable billiard where the billiard table is neither a quadric nor composed of pieces of quadrics.