Skip to main content
SpringerLink
Log in

Rotation Functions of Integrable Billiards As Orbital Invariants

  • MATHEMATICS
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

Orbital invariants of integrable billiards on two-dimensional book tables are studied at constant energy values. These invariants are calculated from rotation functions defined on one-parameter families of Liouville 2-tori. For two-dimensional billiard books, a complete analogue of Liouville’s theorem is proved, action–angle variables are introduced, and rotation functions are defined. A general formula for the rotation functions of such systems is obtained. For a number of examples, the monotonicity of these functions is studied, and edge orbital invariants (rotation vectors) are calculated. It turned out that not all billiards have monotonic rotation functions, as was originally assumed by A. Fomenko’s hypothesis. However, for some series of billiards, this hypothesis is true.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.

Similar content being viewed by others

REFERENCES

  1. A. T. Fomenko and H. Zieschang, “A topological invariant and a criterion for the equivalence of integrable Hamiltonian systems with two degrees of freedom,” Math. USSR-Izv. 36 (3), 567–596 (1991).

    Article  MathSciNet  Google Scholar 

  2. V. V. Fokicheva, “Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas,” Moscow Univ. Math. Bull. 69 (4), 148–158 (2014).

    Article  MathSciNet  Google Scholar 

  3. V. V. Fokicheva, “A topological classification of billiards in locally planar domains bounded by arcs of confocal quadrics,” Sb. Math. 206 (10), 1463–1507 (2015).

    Article  MathSciNet  Google Scholar 

  4. V. V. Vedyushkina and I. S. Kharcheva, “Billiard books model all three-dimensional bifurcations of integrable Hamiltonian systems,” Sb. Math. 209 (12), 1690–1727 (2018).

    Article  MathSciNet  Google Scholar 

  5. V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Mosk. Gos. Univ., Moscow, 1991; Am. Math. Soc., Providence, R.I., 1991).

  6. V. Dragović and M. Radnović, Integrable Billiards, Quadrics, and Multidimensional Poncelet Porisms (Regulyarnaya i Khaoticheskaya Dinamika, Moscow, 2010) [in Russian].

    Google Scholar 

  7. V. Dragović and M. Radnović, “Minkowski plane, confocal conics, and billiards,” Publ. Inst. Math. 94 (108), 17–30 (2013). https://doi.org/10.2298/PIM1308017D

    Article  MathSciNet  Google Scholar 

  8. V. Dragović and M. Radnović, “Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances,” Bull. Am. Math. Soc. 51, 373–445 (2014). https://doi.org/10.1090/S0273-0979-2014-01437-5

    Article  MathSciNet  Google Scholar 

  9. V. Dragović and M. Radnović, “Periods of pseudo-integrable billiards,” Arnold Math. J. 1 (1), 69–73 (2015). https://doi.org/10.1007/s40598-014-0004-0

    Article  MathSciNet  Google Scholar 

  10. V. Dragović and M. Radnović, “Caustics of Poncelet polygons and classical extremal polynomials,” Regular Chaotic Dyn. 24 (1), 1–35 (2019). https://doi.org/10.1134/S1560354719010015

    Article  ADS  MathSciNet  Google Scholar 

  11. A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems: Geometry, Topology, Classification (Udmurt. Univ., Izhevsk, 1999; Chapman and Hall/CRC, Boca Raton, FL, 2004).

  12. V. V. Vedyushkina, “Orbital invariants of flat billiards bounded by arcs of confocal quadrics and containing focuses,” Moscow Univ. Math. Bull. 76 (4), 177–180 (2021).

    Article  MathSciNet  Google Scholar 

  13. V. V. Vedyushkina and A. T. Fomenko, “Billiards and integrability in geometry and physics: New scope and new potential,” Moscow Univ. Math. Bull. 74 (3), 98–107 (2019).

    Article  MathSciNet  Google Scholar 

  14. V. V. Vedyushkina (Fokicheva) and A. T. Fomenko, “Integrable geodesic flows on orientable two-dimensional surfaces and topological billiards,” Izv. Math. 83 (6), 1137–1173 (2019).

    Article  MathSciNet  Google Scholar 

Download references

ACKNOWLEDGMENTS

The authors are grateful to V.V. Vedyushkina and E.A. Kudryavtseva for valuable comments and their interest in this work.

Funding

This work was performed at Lomonosov Moscow State University and was supported by the Russian Science Foundation, project no. 22-71-00111.

Belozerov also acknowledges the support of the Foundation for Advancement of Theoretical Physics and Mathematics “BASIS.”

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    G. V. Belozerov &  A. T. Fomenko

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    A. T. Fomenko

Authors
  1. G. V. Belozerov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. T. Fomenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to G. V. Belozerov or A. T. Fomenko.

Ethics declarations

The authors of this work declare that they have no conflicts of interest.

Additional information

Translated by I. Ruzanova

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Belozerov, G.V., Fomenko, A.T. Rotation Functions of Integrable Billiards As Orbital Invariants. Dokl. Math. (2024). https://doi.org/10.1134/S1064562424701722

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1134/S1064562424701722

Keywords:

Search

Navigation