Classification of symmetric periodic trajectories in ellipsoidal billiards

TL;DR

This paper classifies symmetric periodic trajectories in ellipsoidal billiards across dimensions, providing a complete enumeration, an algorithm for finding minimal trajectories, and illustrating unique 3D properties.

Contribution

It offers a comprehensive classification of symmetric periodic billiard trajectories in ellipsoids and develops an algorithm to identify minimal trajectories in 2D and 3D.

Findings

Exactly 2^{2n}(2^{n+1}-1) classes of trajectories in n-dimensional ellipsoids.

Algorithm successfully finds minimal trajectories in 2D and 3D cases.

Some 3D trajectories exhibit properties not possible in 2D.

Abstract

We classify nonsingular symmetric periodic trajectories (SPTs) of billiards inside ellipsoids of R^{n+1} without any symmetry of revolution. SPTs are defined as periodic trajectories passing through some symmetry set. We prove that there are exactly 2^{2n}(2^{n+1}-1) classes of such trajectories. We have implemented an algorithm to find minimal SPTs of each of the 12 classes in the 2D case (R^2) and each of the 112 classes in the 3D case (R^3). They have periods 3, 4 or 6 in the 2D case; and 4, 5, 6, 8 or 10 in the 3D case. We display a selection of 3D minimal SPTs. Some of them have properties that cannot take place in the 2D case.