### 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 language | English |
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Pages (from-to) | 2706-2710 |

Number of pages | 5 |

Journal | Journal of Mathematical Sciences |

Volume | 128 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 1 2005 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Journal of Mathematical Sciences*,

*128*(2), 2706-2710. https://doi.org/10.1007/s10958-005-0220-1

**Billiards and nonholonomic distributions.** / Baryshnikov, Y.; Zharnitsky, V.

Research output: Contribution to journal › Article

*Journal of Mathematical Sciences*, vol. 128, no. 2, pp. 2706-2710. https://doi.org/10.1007/s10958-005-0220-1

}

TY - JOUR

T1 - Billiards and nonholonomic distributions

AU - Baryshnikov, Y.

AU - Zharnitsky, V.

PY - 2005/7/1

Y1 - 2005/7/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=21544437605&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21544437605&partnerID=8YFLogxK

U2 - 10.1007/s10958-005-0220-1

DO - 10.1007/s10958-005-0220-1

M3 - Article

AN - SCOPUS:21544437605

VL - 128

SP - 2706

EP - 2710

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 2

ER -