Cayley-Type Conditions for Billiards within $k$ Quadrics in $R^d$

TL;DR

This paper derives Cayley-type conditions that characterize periodic billiard trajectories within regions bounded by multiple quadrics in high-dimensional space, extending classical billiard theory to complex geometric configurations.

Contribution

It introduces new notions of reflection and signature for billiard trajectories and derives Cayley-type conditions for periodicity in multi-quadric billiard systems in R^d.

Findings

Cayley-type conditions for periodic billiard trajectories are established.

Conditions for billiards within k quadrics in R^d are formulated.

Limit cases for billiards on a single quadric are described.

Abstract

The notions of reflection from outside, reflection from inside and signature of a billiard trajectory within a quadric are introduced. Cayley-type conditions for periodical trajectories for the billiard in the region bounded by $k$ quadrics in $R^d$ and for the billiard ordered game within $k$ ellipsoids in $R^d$ are derived. In a limit, the condition describing periodic trajectories of billiard systems on a quadric in $R^d$ is obtained.