Non-colliding billiards in the plane

Abstract

We present an open problem about non-colliding freely moving hard disks in the Euclidean plane, together with related positive and negative partial results. The open problem is stated in a non-degenerate form: velocities are required to be pairwise distinct and their speeds are required to be uniformly bounded away from infinity. The positive deterministic result gives a bounded, injective, non-colliding velocity assignment for the integer lattice; after a common velocity shift, the speeds are also bounded away from zero. The negative result shows that no bounded continuous vector field on the whole plane can serve as a universal assignment satisfying the same separation inequality for all pairs of points at distance greater than one. We also record a space-time interpretation of the problem, relate it to packings by nonparallel cylinders in three dimensions, and formulate a corresponding topological-dynamical question for cylinder packings.