Lagrangian relations and linear point billiards

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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. (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.
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  • Office of the Director (DMS-20030177)

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