Billiards and nonholonomic distributions

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Abstract

In this note, we consider billiards with full families of periodic orbits. It is shown that the construction of a convex billiard with a "rational" caustic (i.e., carrying only periodic orbits) can be reformulated as a problem of finding a closed curve tangent to an (N - 1)-dimensional distribution on a (2N - 1)-dimensional manifold. We describe the properties of this distribution, as well as some important consequences for billiards with rational caustics. A very particular application of our construction states that an ellipse can be infinitesimally perturbed so that any chosen rational elliptic caustic will persist.

Original languageEnglish (US)
Pages (from-to)2706-2710
Number of pages5
JournalJournal of Mathematical Sciences
Volume128
Issue number2
DOIs
StatePublished - Jul 2005
Externally publishedYes

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Applied Mathematics

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