Spinning Billiards and Chaos

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.