On Cayley conditions for billiards inside ellipsoids

Billiard trajectories inside an ellipsoid of $\mathbb{R}^n$ are tangent to n − 1 quadrics of the pencil of confocal quadrics determined by the ellipsoid. The quadrics associated with periodic trajectories verify certain algebraic conditions. Cayley found them for the planar case. Dragović and Radnović generalized them to any dimension. We rewrite the original matrix formulation of these generalized Cayley conditions as a simpler polynomial one. We find several algebraic relations between caustic parameters and ellipsoidal parameters that give rise to non-singular periodic trajectories. These relations become remarkably simple when the elliptic period is minimal. We study the caustic types, the winding numbers and the ellipsoids of such minimal periodic trajectories. We also describe some non-minimal periodic trajectories.