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On fourth-degree polynomial integrals of the Birkhoff billiard

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Abstract

We study the Birkhoff billiard in a convex domain with a smooth boundary γ. We show that if this dynamical system has an integral which is polynomial in velocities of degree 4 and is independent with the velocity norm, then γ is an ellipse.

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Authors and Affiliations

  1. Tel-Aviv University, Tel Aviv, Israel

    M. Bialy

  2. Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia

    A. E. Mironov

Authors
  1. M. Bialy
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  2. A. E. Mironov
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Corresponding author

Correspondence to M. Bialy.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 34–40.

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Bialy, M., Mironov, A.E. On fourth-degree polynomial integrals of the Birkhoff billiard. Proc. Steklov Inst. Math. 295, 27–32 (2016). https://doi.org/10.1134/S0081543816080022

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  • DOI: https://doi.org/10.1134/S0081543816080022

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