Sub-riemannian geometry and periodic orbits in classical billiards

Yuliy Baryshnikov, Vadim Zharnitsky

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)

Abstract

Classical (Birkhoff) billiards with full 1-parameter families of periodic orbits are considered. It is shown that construction of a convex billiard with a "rational" caustic (i.e. carrying only periodic orbits) can be reformulated as the problem of finding a closed curve tangent to a non-integrable distribution on a manifold. The properties of this distribution are described as well as the consequences for the billiards with rational caustics. A particular implication of this construction is that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist.

Original languageEnglish
Pages (from-to)587-598
Number of pages12
JournalMathematical Research Letters
Volume13
Issue number4
DOIs
Publication statusPublished - Jul 2006

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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