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Linear point billiards, Lagrangian relations and the high-energy limit of the N-body problem


por el

Dr. Richard Montgomery

Department of Mathematics, University of California Santa Cruz

Resumen:
Motivated by trying to understand the high-energy limit of the N-body problem we will construct non-deterministic ``linear billiard processes'' whose data consists of a finite collection of linear subspaces of a Euclidean vector space. The trajectories of the process are piecewise linear constant speed curves which may only change direction upon colliding with one of the subspaces, at which instant their change of direction is subject to the standard law of reflection for billiards. (This ``dynamics'' is nondeterministic whenever a subspace has codimension c greater than 1 since for a given incoming ray, the reflection law only asserts that the outgoing ray's velocity lies on a certain $c-1$-dimensional sphere.) The ``itinerary'' of a trajectory is the list of subspaces it hits, in the order hit. Two basic questions are: (A) Are itineraries always finite? (B) What is the structure of the space of all trajectories having a fixed itinerary? In a beautiful paper Burago-Ferleger-Kononenko [BFK] used metric geometry ideas and the notion of a ``Hadamard space'' to answer (A) affirmatively. We answer (B): this space of trajectories forms a Lagrangian relation on the space of lines in $E$. Our proof relies on two techniques, (1) generating families for Lagrangian relations, and (2) the metric geometry constructions of BFK which rely crucially on a theorem of Reshetynak. This is joint work with Andreas Knauf and Jacques Fejoz.

MiƩrcoles 13 de abril, 2016
17:00 hrs.
(Notar que el coloquio es una hora antes de lo normal.)
Salón 203, Edificio Anexo, IIMAS

Informes: coloquiomym@gmail.com, o al 5622-3564.