Spinning Billiards and Chaos

TL;DR

This paper explores how internal spin affects chaos in billiard systems, showing that spin reduces but does not eliminate chaos, with effects depending on geometry and a conserved quantity influencing dynamics.

Contribution

It introduces a model coupling translational and rotational motion in billiards, revealing how spin suppresses chaos and identifying a conserved quantity affecting phase space structure.

Findings

Spin reduces the Lyapunov exponent in chaotic billiards.

In geometries with flat sections, spin creates islands of regularity.

The classical chaos scaling law fails for spinning billiards.

Abstract

We investigate the impact of internal spin on chaos in billiard systems. Extending the standard point-particle billiard by coupling translational and rotational degrees of freedom through a dimensionless spin parameter $\alpha = I/(mr^2) \in [0,1]$, we find that spin reduces chaos monotonically but does not eliminate it. In the Bunimovich stadium and Sinai billiard, the Lyapunov exponent decreases with $\alpha$ but remains positive throughout the physical range, while the circle and rectangle remain integrable. Finite-time Lyapunov exponent distributions reveal a mixed phase space in which spin creates islands of regularity while the majority of trajectories remain chaotic. The mechanism is a conserved quantity $Q = v_\parallel - \alpha u$ preserved through each collision, which constrains the dynamics on sequences of same-orientation wall collisions and explains why spin suppresses chaos more effectively in geometries with longer flat sections. We further show that the Datseris--Hupe--Fleischmann scaling $\lambda \propto 1/f_{\rm chaotic}$ fails for spinning billiards: spin reduces the intensity of chaos, not merely the fraction of chaotic trajectories.