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Linear billiard systems, the positive-energy N-body problem and Lagrangian relations Montgomery, Richard


Recall Rutherford scattering. A 2-body problem was solved with the key ingredient being scattering: incoming and outgoing lines which were asymptotes to associated hyperbolae. In the N-body problem at positive energies, when a certain limit is taken, families of solutions converge to a system of N incoming and N outgoing rays with a finite number of elastic collisions in between. We abstract this situation to a ``point billiard process'' whose data consist of a Euclidean vector space E endowed with a finite collection of codimension d linear subspaces called ``collision subspaces''. The solutions to the `process'' are unit speed continuous piecewise linear trajectories corresponding to billiards played on the table E minus the union of the collision subspaces. These solutions, or ``billiard trajectories'' move in straight lines away from the collision subspaces. Upon hitting a subspace the reflect off according to the standard law of reflection. The ``dynamics'' associated to the process is not deterministic since for a given ray incoming to a collision subspace there is a $d-1$ dimensional sphere's worth of allowable outgoing rays. The ``itinerary'' of such a trajectory is the list of subspaces it hits, in the order hit. Two basic questions are: (A) Are itineraries finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In two beautiful papers from 1998 Burago-Ferleger-Kononenko [BFK] use non-smooth metric geometry ideas (CAT(0) and Hadamard spaces) to answer (A) affirmatively. We answer (B): this space of trajectories is a Lagrangian relation on the space of oriented lines in E. Our proof relies on two techniques, (1) generating families for Lagrangian relations, and (2) the metric geometry introduced by BFK and relying crucially on a theorem of Reshetynak. We will focus on (2). This is joint work with Andreas Knauf and Jacques Fejoz.

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