Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics. (English) Zbl 1252.37049
In this paper the authors study the geometry of confocal quadrics in pseudo-Euclidean spaces of an arbitrary dimension \(d\) and any signature, and related billiard dynamics. The goal is to give a complete description of periodic billiard trajectories within ellipsoids. The novelty of their approach is based on the introduction of a new discrete combinatorial-geometric structure associated to a confocal pencil of quadrics, a colouring in \(d\) colours, by which they decompose quadrics of \(d+1\) geometric types of a pencil into new relativistic quadrics of \(d\) relativistic types. Deep insight of related geometry and combinatorics comes from their study of what they call discriminant sets of tropical lines \(\Sigma^+\) and \(\Sigma^-\) and their singularities. All this allows them to get an analytic criterion describing all periodic billiard trajectories, including the light-like ones as those of a special interest.
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
Keywords:
confocal quadrics; Poncelet theorem; periodic billiard trajectories; Minkowski space; light-like billiard trajectories; tropic curvesReferences:
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