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：我们通过在拓扑克林场论框架内编码其动力学，证明二维台球系统是图灵完备的。台球系统作为具有弹性反射的粒子运动理想化模型，在陡峭约束势下作为光滑哈密顿系统的极限自然产生。我们的研究结果确立了物理自然台球类模型中不可判定轨迹的存在性，包括硬球气体中产生的台球类模型及天体力学碰撞链极限中的台球类模型。