Lagrangian relations and linear point billiards. (English) Zbl 1375.37114
Authors’ 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 D. Burago et al. [Ann. Math. (2) 147, No. 3, 695–708 (1998; Zbl 0995.37025); Ergodic Theory Dyn. Syst. 18, No. 2, 303–319 (1998; Zbl 0915.58057); A course in metric geometry. Providence, RI: American Mathematical Society (AMS) (2001; Zbl 0981.51016)] 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.
Reviewer: Maria Gousidou-Koutita (Thessaloniki)
MSC:
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 70F10 | \(n\)-body problems |
| 81U10 | \(n\)-body potential quantum scattering theory |
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