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Tr. Mat. Inst. Steklova, 2016, Volume 295, Pages 34–40 (Mi tm3748)  

This article is cited in 4 scientific papers (total in 4 papers)

On fourth-degree polynomial integrals of the Birkhoff billiard

M. Bialya, A. E. Mironovb

a Tel-Aviv University, Tel Aviv, Israel
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090 Russia

Abstract: We study the Birkhoff billiard in a convex domain with a smooth boundary $\gamma$. 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 $\gamma$ is an ellipse.

Funding Agency Grant Number
Israel Science Foundation 162/15
Russian Science Foundation 14-11-00441
The work of M. Bialy is supported by the Israel Science Foundation under grant 162/15. The work of A. E. Mironov is supported by the Russian Science Foundation under grant 14-11-00441.


DOI: https://doi.org/10.1134/S0371968516040026

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 295, 27–32

Bibliographic databases:

UDC: 531.01
Received: June 14, 2016

Citation: M. Bialy, A. E. Mironov, “On fourth-degree polynomial integrals of the Birkhoff billiard”, Modern problems of mechanics, Collected papers, Tr. Mat. Inst. Steklova, 295, MAIK Nauka/Interperiodica, Moscow, 2016, 34–40; Proc. Steklov Inst. Math., 295 (2016), 27–32

Citation in format AMSBIB
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\pages 34--40
\publ MAIK Nauka/Interperiodica
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. Bialy, A. E. Mironov, “A survey on polynomial in momenta integrals for billiard problems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170418  crossref  isi  scopus
    2. A. A. Glutsyuk, “On two-dimensional polynomially integrable billiards on surfaces of constant curvature”, Dokl. Math., 98:1 (2018), 382–385  crossref  crossref  zmath  isi  elib  scopus
    3. A. Glutsyuk, E. Shustin, “On polynomially integrable planar outer billiards and curves with symmetry property”, Math. Ann., 372:3-4 (2018), 1481–1501  crossref  mathscinet  zmath  isi
    4. M. Byalyi, A. E. Mironov, “Polinomialnaya neintegriruemost magnitnykh bilyardov na sfere i giperbolicheskoi ploskosti”, UMN, 74:2(446) (2019), 3–26  mathnet  crossref  elib
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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