Classical billiards can compute

Mathematics > Dynamical Systems [Submitted on 22 Dec 2025] Title:Classical billiards can compute View PDF HTML (experimental) Abstract: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. Current browse context: math.DS References & Citations export BibTeX citation Loading... Bibliographic and Citation Tools Bibliographic Explorer ( What is the Explorer? ) Connected Papers ( What is Connected Papers? ) Litmaps ( What is Litmaps? ) scite Smart Citations ( What are Smart Citations? ) Code, Data and Media Associated with this Article alphaXiv ( What is alphaXiv? ) CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub ( What is DagsHub? ) Gotit.pub ( What is GotitPub? ) Hugging Face ( What is Huggingface? ) Papers with Code ( What is Papers with Code? ) ScienceCast ( What is ScienceCast? ) Demos Recommenders and Search Tools Influence Flower ( What are Influence Flowers? ) CORE Recommender ( What is CORE? ) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

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