No-slip Billiards with Particles of Variable Mass Distribution

TL;DR

This paper explores how varying the mass distribution of particles in no-slip billiards significantly alters their dynamical behavior, revealing a spectrum from regular to chaotic dynamics and unusual phenomena like full measure periodic orbits.

Contribution

It introduces new families of no-slip billiards with variable mass distributions, demonstrating diverse and novel dynamical behaviors not seen in standard billiards.

Findings

Diverse dynamics from integrable to chaotic observed.

Existence of full measure periodic billiards in certain cases.

Mass distribution changes induce significant dynamical shifts.

Abstract

Astute variations in the geometry of mathematical billiard tables have been and continue to be a source of understanding their wide range of dynamical behaviors, from regular to chaotic. Viewing standard specular billiards in the broader setting of no-slip (or rough) collisions, we show that an equally rich spectrum of dynamics can be called forth by varying the mass distribution of the colliding particle. We look at three two-parameter families of billiards varying both the geometry of the table and the particle, including as special cases examples of standard billiards demonstrating dynamics from integrable to chaotic, and show that markedly divergent dynamics may arise by changing only the mass distribution. Furthermore, for certain parameters billiards emerge which display unusual dynamics, including examples of full measure periodic billiards, conjectured to be nonexistent for the standard billiards in Euclidean domains.