Upper bound on the number of collisions of pinned billiard balls

TL;DR

This paper establishes an explicit upper bound on the maximum number of pseudo-collisions in a system of fixed-position balls with changing pseudo-velocities, using folding analysis in a d-dimensional space.

Contribution

It introduces a novel approach to bounding collision counts in pinned ball systems through folding mappings and orbit size analysis.

Findings

Derived an explicit upper bound for pseudo-collisions in pinned ball systems.

Applied folding analysis to formalize collision dynamics.

Provided mathematical proof for the bound in arbitrary dimensions.

Abstract

We consider systems of "pinned balls," i.e., balls that have fixed positions and pseudo-velocities. Pseudo-velocities change according to the same rules as those for velocities of totally elastic collisions between moving balls. The times of possible pseudo-collisions for different pairs of pinned balls are chosen in an exogenous way. We give an explicit upper bound for the maximum number of pseudo-collisions for a system of $n$ pinned balls in a $d$-dimensional space. The proof is based on analysis of foldings, i.e., mappings that formalize the idea of folding a piece of paper along a crease. We prove an upper bound for the size of an orbit of a point subjected to foldings.