We show that two-dimensional billiard systems are Turing complete by encoding their dynamics within the framework of Topological Kleene Field Theory. 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：我们通过将二维台球系统的动力学编码在拓扑克林场论的框架内，证明了此类系统具有图灵完备性。台球系统作为粒子弹性反射运动的理想化模型，自然地出现在陡峭约束势下光滑哈密顿系统的极限情形中。我们的研究结果确立了在物理上自然的台球类模型（包括硬球气体中出现的台球类模型及天体力学碰撞链极限中的台球类模型）中存在不可判定的运动轨迹。