Polynomial non-integrability of magnetic billiards on the sphere and the hyperbolic plane

M. Bialy, A. E. Mironov

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3 Citations (Scopus)


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 languageEnglish
Article number1
Pages (from-to)187-209
Number of pages23
JournalRussian Mathematical Surveys
Issue number2
Publication statusPublished - 16 Jan 2019


  • constant-curvature surfaces
  • Magnetic billiards
  • polynomial integrals
  • magnetic billiards

State classification of scientific and technological information

  • 29 PHYSICS


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