Rolling systems and their billiard limits

TL;DR

This paper explores the relationship between no-slip billiard systems and non-holonomic rolling systems, demonstrating that no-slip billiards can be viewed as limits of rolling systems, similar to how traditional billiards relate to geodesic flows.

Contribution

It establishes a theoretical connection showing no-slip billiards as limits of non-holonomic rolling systems, extending the understanding of billiard models in mechanical systems.

Findings

No-slip billiards arise as limits of non-holonomic rolling systems.

The relationship parallels how billiards relate to geodesic flows.

Provides a new perspective on the modeling of mechanical interactions.

Abstract

Billiard systems, broadly speaking, may be regarded as models of mechanical systems in which rigid parts interact through elastic impulsive (collision) forces. When it is desired or necessary to account for linear/angular momentum exchange in collisions involving a spherical body, a type of billiard system often referred to as no-slip has been used. In recent work, it has become apparent that no-slip billiards resemble non-holonomic mechanical systems in a number of ways. Based on an idea by Borisov, Kilin and Mamaev, we show that no-slip billiards very generally arise as limits of non-holonomic (rolling) systems, in a way that is akin to how ordinary billiards arise as limits of geodesic flows through a flattening of the Riemannian manifold.