### 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 language | English |
---|---|

Pages (from-to) | 587-598 |

Number of pages | 12 |

Journal | Mathematical Research Letters |

Volume | 13 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jul 2006 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematical Research Letters*,

*13*(4), 587-598. https://doi.org/10.4310/MRL.2006.v13.n4.a8

**Sub-riemannian geometry and periodic orbits in classical billiards.** / Baryshnikov, Yuliy; Zharnitsky, Vadim.

Research output: Contribution to journal › Article

*Mathematical Research Letters*, vol. 13, no. 4, pp. 587-598. https://doi.org/10.4310/MRL.2006.v13.n4.a8

}

TY - JOUR

T1 - Sub-riemannian geometry and periodic orbits in classical billiards

AU - Baryshnikov, Yuliy

AU - Zharnitsky, Vadim

PY - 2006/7

Y1 - 2006/7

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33748921398&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33748921398&partnerID=8YFLogxK

U2 - 10.4310/MRL.2006.v13.n4.a8

DO - 10.4310/MRL.2006.v13.n4.a8

M3 - Article

AN - SCOPUS:33748921398

VL - 13

SP - 587

EP - 598

JO - Mathematical Research Letters

JF - Mathematical Research Letters

SN - 1073-2780

IS - 4

ER -