Billiard ball in the space with a time machine

We study a system of an elastic ball moving in the nonrelativistic spacetime with a nontrivial causal structure produced by a wormhole-based time machine. For such a system, it is possible to formulate a simple model of the so-called "grandfather paradox": for certain "paradoxical" initial conditions, the standard straight trajectory of the ball would self-collide inconsistently. We analyze globally consistent solutions of local equations of motion; namely, we find all trajectories with one self-collision. It is demonstrated that all standard initial conditions have a consistent evolution, including those paradoxical ones, for which the inconsistent collision-free trajectory is superseded by a special consistent self-colliding trajectory. Moreover, it is shown that for a wide class of initial conditions, more than one globally consistent evolution exist. The nontrivial causal structure thus breaks the uniqueness of the classical theory even for locally deterministic physical laws.