标题： 关于旋转台球和混沌的一点注记 显示英文标题

标题： A Note on Spinning Billiards and Chaos

摘要： 我们研究了内部自由度——特别是自旋——对台球系统经典动力学的影响。尽管传统研究将台球建模为经历镜面反射的点粒子，我们将这一范式扩展到有限尺寸效应和角动量，并引入了一个无量纲的自旋参数$\alpha$来表征惯性矩。通过在圆形、矩形、椭圆和 Sinai 几何结构下的数值模拟，我们分析了由此产生的轨迹，并通过最大李亚普诺夫指数量化混沌程度。令人惊讶的是，我们发现即使在经典的混沌几何结构中，自旋也能使动力学规律化：对于广泛的$\alpha$范围，在椭圆和 Sinai 桌面上，李亚普诺夫指数在晚期时间消失，表明混沌被抑制。这一效果通过相空间分析得到了证实，该分析显示了邻近轨迹的非指数发散。我们的结果表明，内部结构可以定性地改变系统的动力学图景，可能作为一种更广泛背景下的混沌抑制机制。 显示英文摘要

摘要： We investigate the impact of internal degrees of freedom - specifically spin - on the classical dynamics of billiard systems. While traditional studies model billiards as point particles undergoing specular reflection, we extend the paradigm by incorporating finite-size effects and angular momentum, introducing a dimensionless spin parameter $\alpha$ that characterizes the moment of inertia. Using numerical simulations across circular, rectangular, stadium, and Sinai geometries, we analyze the resulting trajectories and quantify chaos via the leading Lyapunov exponent. Strikingly, we find that spin regularizes the dynamics even in geometries that are classically chaotic: for a wide range of $\alpha$, the Lyapunov exponent vanishes at late times in the stadium and Sinai tables, signaling suppression of chaos. This effect is corroborated by phase space analysis showing non-exponential divergence of nearby trajectories. Our results suggest that internal structure can qualitatively alter the dynamical landscape of a system, potentially serving as a mechanism for chaos suppression in broader contexts.