Billiard dynamics of bouncing dumbbell

TL;DR

This paper analyzes the billiard dynamics of a dumbbell-shaped system with two masses connected by a rod, establishing bounds on boundary collisions and deriving an adiabatic invariant under specific conditions.

Contribution

It introduces new bounds on the number of boundary collisions and derives an adiabatic invariant for fast-rotating, large-mass dumbbells using billiard techniques.

Findings

Bound on the number of boundary collisions established

Adiabatic invariant derived for specific mass ratios and rotation speeds

Billiard techniques applied to a dumbbell system

Abstract

A system of two masses connected with a weightless rod (called dumbbell in this paper) interacting with a flat boundary is considered. The sharp bound on the number of collisions with the boundary is found using billiard techniques. In case, the ratio of masses is large and the dumbbell rotates fast, an adiabatic invariant is obtained.