We show that two-dimensional billiard systems are Turing complete, in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a specified open set. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems under steep confining potentials. Our results establish the existence of undecidable trajectories in physically natural billiard-type models, including billiard-type models arising in hard-sphere gases and in collision-chain limits of celestial mechanics.

翻译：我们证明二维台球系统是图灵完备的，即任意图灵机在给定输入下的停机问题等价于该系统中特定有界轨迹进入指定开集的问题。台球系统是弹性反射粒子运动的理想化模型，并自然作为陡峭约束势下光滑哈密顿系统的极限而出现。我们的结果确立了物理上自然的台球型模型中存在不可判定轨迹，包括从硬球气体和天体力学碰撞链极限中导出的台球型模型。