  We study a system of an elastic ball moving in the non-relativistic 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superseded by a special consistent self-colliding trajectory.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.
