## Abstract

Magnetic billiards in a convex domain with smooth boundary on a constant-curvature surface in a constant magnetic feld is considered in this paper. The question of the existence of an integral of motion which is a polynomial in the components of the velocity is investigated. It is shown that if such an integral exists, then the boundary of the domain defnes a non-singular algebraic curve in C3. It is also shown that for a domain other than a geodesic disk, magnetic billiards does not admit a polynomial integral for all but perhaps fnitely many values of the magnitude of the magnetic feld. To prove our main theorems a new dynamical system, outer magnetic billiards , on a constant-curvature surface is introduced, a system dual to magnetic billiards. By passing to this dynamical system one can apply methods of algebraic geometry to magnetic billiards.

Original language | English |
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Article number | 1 |

Pages (from-to) | 187-209 |

Number of pages | 23 |

Journal | Russian Mathematical Surveys |

Volume | 74 |

Issue number | 2 |

DOIs | |

Publication status | Published - 16 Jan 2019 |

## Keywords

- constant-curvature surfaces
- Magnetic billiards
- polynomial integrals
- CLASSICAL BILLIARDS
- magnetic billiards
- INTEGRABLE BILLIARDS
- SURFACES

## State classification of scientific and technological information

- 29 PHYSICS