Classical billiards can compute

TL;DR

This paper proves that two-dimensional billiard systems are Turing complete, meaning they can simulate any computation, including undecidable problems, in physically natural models like gases and celestial mechanics.

Contribution

It establishes the existence of undecidable trajectories in natural billiard models, linking computational theory with physical dynamical systems.

Findings

Billiard systems can simulate Turing machines.

Undecidable trajectories exist in natural billiard models.

Results connect computational theory with physical particle dynamics.

Abstract

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.