On Cayley conditions for billiards inside ellipsoids

TL;DR

This paper simplifies the algebraic conditions characterizing periodic billiard trajectories inside ellipsoids in any dimension, revealing new relations between parameters that lead to nonsingular periodic paths.

Contribution

It reformulates Cayley's conditions from a matrix to a polynomial form and uncovers algebraic relations linking caustic and ellipsoidal parameters for periodic trajectories.

Findings

Simplified polynomial formulation of Cayley conditions.

Identified algebraic relations for nonsingular periodic trajectories.

Extended known planar results to higher dimensions.

Abstract

All the segments (or their continuations) of a billiard trajectory inside an ellipsoid of $\Rset^n$ are tangent to n-1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated to periodic billiard trajectories verify certain algebraic conditions. Cayley found them in the planar case. Dragovi\'{c} and Radnovi\'{c} generalized them to any dimension. We rewrite the original matrix formulation of these generalized Cayley conditions as a simpler polynomial one. We find several remarkable algebraic relations between caustic parameters and ellipsoidal parameters that give rise to nonsingular periodic trajectories.