Spinning Billiards and Chaos

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 $α= I/(mr^2) \in [0,1]$, we find that spin reduces chaos monotonically but does not eliminate it. In the Bunimovich stadium and Sinai billiard, the Lyapunov exponent decreases with $α$ 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 $Q = v_\parallel - αu$ 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 $λ\propto 1/f_{\rm chaotic}$ fails for spinning billiards: spin reduces the intensity of chaos, not merely the fraction of chaotic trajectories.

Figures (14)

• Figure 1: Representative trajectories (500 collisions) for all four geometries at $\alpha = 0$, $0.3$, $0.7$, and $1.0$, colored from early (dark) to late (light) collisions. Spin coupling visibly modifies trajectory structure, creating envelope patterns and apparent regularization, but the stadium and Sinai remain chaotic by the Lyapunov criterion.
• Figure 2: Ensemble-averaged Lyapunov characteristic number as a function of spin parameter $\alpha$ for all four geometries (50 values of $\alpha$, $10^5$ ICs, $5\times 10^4$ collisions each for the stadium and Sinai). Error bars show the standard error of the mean. The circle and rectangle remain integrable (LCN $\approx 0$), while the stadium and Sinai show monotonically decreasing but persistently positive LCN.
• Figure 3: Finite-time Lyapunov exponent distributions for the stadium billiard at six values of $\alpha$, computed from $10^5$ random initial conditions ($5\times 10^4$ collisions each). Vertical lines show the mean (black, solid), median (blue, dashed), and conditional mean of the chaotic component (green, dotted). The unimodal distribution at $\alpha=0$ becomes bimodal for $\alpha \gtrsim 0.3$, with a secondary peak near zero indicating regular trajectories.
• Figure 4: Chaotic fraction (percentage of initial conditions with FTLE $> 0.01$) as a function of $\alpha$ for the stadium and Sinai geometries ($10^5$ ICs, $5\times 10^4$ collisions each, 50 values of $\alpha$). The shaded bands show the range obtained by varying the FTLE threshold between $0.001$ and $0.05$; the near-invisibility of the bands confirms robustness to the threshold choice.
• Figure 5: Phase space separation $\ln(\delta_n/\delta_0)$ vs collision number for the four geometries at several $\alpha$ values ($8{,}000$ trajectory pairs, $\delta_0 = 10^{-7}$, 50 collisions). Linear growth (stadium, Sinai) indicates exponential divergence and genuine chaos; logarithmic growth (circle, rectangle) indicates integrable dynamics.