Skip to main content

On Two-Dimensional Polynomially Integrable Billiards on Surfaces of Constant Curvature

Doklady Mathematics volume 98, pages 382–385 (2018)Cite this article

Abstract

The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewise C2-smooth boundary on surfaces of constant curvature: Euclidean plane, sphere, and Lobachevsky plane. Namely, we obtain a complete classification of billiards for which the billiard geodesic flow has a nontrivial first integral depending polynomially on the velocity. According to this classification, every polynomially integrable convex bounded planar billiard with C2-smooth boundary is an ellipse. This is a joint result of M. Bialy, A.E. Mironov, and the author. The proof consists of two parts. The first part was given by Bialy and Mironov in their two joint papers, where the result was reduced to an algebraic-geometric problem, which was partially studied there. The second part is the complete solution of the algebraic-geometric problem presented below.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. A. M. Abdrakhmanov, Moscow Univ. Mech. Bull. 45 (6), 13–17 (1990).

    MathSciNet  Google Scholar 

  2. A. M. Abrdrakhmanov, Moscow Univ. Mech. Bull. 45 (5), 14–16 (1990).

    Google Scholar 

  3. A. Avila, J. De Simoi, and V. Kaloshin, Ann. Math. 184, 527–558 (2016).

    Article  MathSciNet  Google Scholar 

  4. M. Bialy and A. Mironov, Adv. Math. 313, 102–126 (2017).

    Article  MathSciNet  Google Scholar 

  5. M. Bialy and A. E. Mironov, Geom. Phys. 115, 150–156 (2017).

    Article  MathSciNet  Google Scholar 

  6. M. Bialy and A. E. Mironov, Proc. Steklov Inst. Math. 295, 27–32 (2016).

    Article  Google Scholar 

  7. S. V. Bolotin, Moscow Univ. Mech. Bull. 45 (2), 10–13 (1990).

    Google Scholar 

  8. S. V. Bolotin, Math. Notes 51 (2), 117–123 (1992).

    Article  MathSciNet  Google Scholar 

  9. V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts (Mosk. Gos. Univ., Moscow, 1991); Transl. Math. Monographs 89, Am. Math. Soc., Providence, RI, 1991.

    Book  MATH  Google Scholar 

  10. A. Ramani, A. Kalliterakis, B. Grammaticos, and B. Dorizzi, Phys. Lett. A 115 (1–2), 13–17 (1986).

    MathSciNet  Google Scholar 

  11. S. Tabachnikov, Pacific J. Math. 235 (1), 101–104 (2008).

    Article  Google Scholar 

  12. A. P. Veselov, Funct. Anal. Appl. 22 (2), 83–93 (1988).

    Article  MathSciNet  Google Scholar 

  13. A. P. Veselov, J. Geom. Phys. 7 (1), 81–107 (1990).

    Article  MathSciNet  Google Scholar 

  14. V. Kaloshin and A. Sorrentino, Ann. Math. 188 (1), 315–380 (2018).

    Article  MathSciNet  Google Scholar 

  15. A. Glutsyuk and E. Shustin, Math. Ann. (to appear). https://doi.org/10.1007/s00208-018-1726-4.

Download references

Author information

Authors and Affiliations

  1. CNRS, France (UMR 5669 (UMPA, ENS de Lyon) and UMI 2615 (Interdisciplinary Scientific Center J.-V. Poncelet)) National Research University Higher School of Economics, Moscow, Russia

    A. A. Glutsyuk

Authors
  1. A. A. Glutsyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Glutsyuk.

Additional information

Original Russian Text © A.A. Glutsyuk, 2018, published in Doklady Akademii Nauk, 2018, Vol. 481, No. 6.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Glutsyuk, A.A. On Two-Dimensional Polynomially Integrable Billiards on Surfaces of Constant Curvature. Dokl. Math. 98, 382–385 (2018). https://doi.org/10.1134/S1064562418050253

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562418050253