  Motivated by the high-energy limit of the $N$-body problem we construct
non-deterministic billiard process. The billiard table is the complement of a
finite collection of linear subspaces within a Euclidean vector space. A
trajectory is a constant speed polygonal curve with vertices on the subspaces
and change of direction upon hitting a subspace governed by `conservation of
momentum' (mirror reflection). The itinerary of a trajectory is the list of
subspaces it hits, in order. Two basic questions are: (A) Are itineraries
finite? (B) What is the structure of the space of all trajectories having a
fixed itinerary? In a beautiful series of papers Burago-Ferleger-Kononenko
[BFK] answered (A) affirmatively by using non-smooth metric geometry ideas and
the notion of a Hadamard space. We answer (B) by proving that this space of
trajectories is diffeomorphic to a Lagrangian relation on the space of lines in
the Euclidean space. Our methods combine those of BFK with the notion of a
generating family for a Lagrangian relation.
