Counting Collisions in an $N$-Billiard System Using Angles Between Collision Subspaces

TL;DR

This paper calculates principal angles between collision subspaces in an N-billiard system, providing bounds on the number of collisions, and compares these bounds across different dimensions.

Contribution

It introduces a method to compute principal angles for both equal and arbitrary masses, and derives collision bounds for the planar 3-billiard system.

Findings

Principal angles between collision subspaces are explicitly computed.

A bound on the number of collisions in the planar 3-billiard system is established.

Comparison with known bounds in lower dimensions is discussed.

Abstract

The principal angles between binary collision subspaces in an $N$-billiard system in $d$-dimensional Euclidean space are computed. These angles are computed for equal masses and arbitrary masses. We then provide a bound on the number of collisions in the planar 3-billiard system problem. Comparison of this result with known billiard collision bounds in lower dimensions is discussed.