Lagrangian relations and linear point billiards

Féjoz J, Knauf A, Montgomery R (2017)

Publication Status: Published

Publication Type: Journal article

Publication year: 2017



Book Volume: 30

Pages Range: 1326-1355

Journal Issue: 4

DOI: 10.1088/1361-6544/aa5b26


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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Féjoz, J., Knauf, A., & Montgomery, R. (2017). Lagrangian relations and linear point billiards. Nonlinearity, 30(4), 1326-1355.


Féjoz, Jacques, Andreas Knauf, and Richard Montgomery. "Lagrangian relations and linear point billiards." Nonlinearity 30.4 (2017): 1326-1355.

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