Lagrangian Relations and Linear Point Billiards

TL;DR

This paper studies a non-deterministic billiard process in Euclidean spaces with linear subspace boundaries, characterizing the space of trajectories as a Lagrangian relation, extending previous geometric results.

Contribution

It introduces a novel description of the trajectory space as a Lagrangian relation, building on and extending prior geometric analyses of billiard dynamics.

Findings

The trajectory space is diffeomorphic to a Lagrangian relation.

Established a connection between billiard trajectories and Lagrangian relations.

Extended previous geometric results to a broader class of billiard problems.

Abstract

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.