Skip to main content
SpringerLink
Log in

Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas

  • Published:
Moscow University Mathematics Bulletin Aims and scope

Abstract

An integrable system, namely, a billiard in a domain bounded by confocal ellipses and hyperbolas is studied. Such system arises in description of the motion of a point inside this domain with a natural reflection rule at the boundary. The topological invariant of the Liouville equivalence of such systems, namely, the Fomenko-Zieschang molecule, is calculated using a new method developed by the author.

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

Similar content being viewed by others

References

  1. V. V. Kozlov and D. V. Treshchev, Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts (Moscow State Univ., Moscow, 1991; in series: Translations of Math. Monographs, vol. 89, Airier. Math. Soc., Providence, RI: 1991).

    Google Scholar 

  2. A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology and Classification (Publ. House “Udmurtskii Universitet,” Izhevsk, 1999; CRC Press, Boca Raton, FL, 2004).

    Google Scholar 

  3. G. D. Birkhoff, Dynamical Systems (AMS, Providence, RI, 1927; Publ. House “Udmurtskii Universitet,” Izhevsk, 1999).

    MATH  Google Scholar 

  4. A. V. Bolsinov and A. T. Fomenko, “Orbital Classification of Geodesic Flows on Two-Dimensional Ellipsoids. The Jacobi Problem is Orbitally Equivalent to the Integrable Euler Case in Rigid Body Dynamics,” Func. Analiz i Ego Prilozh. 29(3), 1 (1995) [Funct. Anal. Appl. 29 (3), 149 (1995)].

    MathSciNet  Google Scholar 

  5. V. Dragovic and M. Radnovic, “Bifurcations of Liouville Tori in Elliptical Billiards,” Regul. Chaotic Dyn. 14(4–5), 479 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  6. V. Dragovic and M. Radnovic, Integrable Billiards, Quadrics, and Multidimensional Poncelet Porisms (NITs “Regular and Chaotic Dynamics,” Moscow, Izhevsk, 2010; Poncelet Porisms and Beyond Springer Birkhäser, Basel, 2011].

    Google Scholar 

  7. V. I. Arnold, Mathematical Methods of Classical Mechanics (Nauka, Moscow, 1989; Springer-Verlag, Heidelberg, N.Y., 1989).

    Book  Google Scholar 

  8. A. T. Fomenko and H. Zieschang, “On Typical Topological Properties of Integrable Hamiltonian Systems,” Izvestiya Akad. Nauk SSSR, Ser. Matem. 52(2), 378 (1988) [Mathematics of the USSR — Izvestiya 32 (2), 385 (1989)].

    MATH  MathSciNet  Google Scholar 

  9. A. T. Fomenko, “The Symplectic Topology of Completely Integrable Hamiltonian Systems,” Uspekhi Matem. Nauk 44(1), 145 (1989) [Russian Math. Surveys 44 (1), 181 (1989)].

    MATH  MathSciNet  Google Scholar 

  10. A. T. Fomenko and H. Zieschang, “A Topological Invariant and a Criterion for the Equivalence of Integrable Hamiltonian Systems with Two Degrees of Freedom,” Izvestiya Akad. Nauk SSSR, Ser. Matem. 54(3), 546 (1990) [Mathematics of the USSR — Izvestiya 36 (3), 567 (1991)].

    MATH  MathSciNet  Google Scholar 

  11. E. Gutkin, “Billiard Dynamics: a Survey with the Emphasis on Open Problems,” Regul. Chaotic Dyn. 8(1), 1 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  12. V. V. Fokicheva, “Description of Singularities for System “Billiard in an Ellipse,” Vestnik Mosk. Univ., Matem. Mekhan., №5, 31 (2012) [Moscow Univ. Math. Bull. 67 (5–6), 217 (2012)].

    Google Scholar 

  13. A. T. Fomenko, E. A. Kudryavtseva, and I. M. Nikonov, “Maximally Symmetric Cell Decompositions of Surfaces and Their Coverings,” Matem. Sbornik 199(9), 3 (2008) [Sbornik Math. 199 (9), 1263 (2008)].

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119991, Russia

    V. V. Fokicheva

Authors
  1. V. V. Fokicheva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Fokicheva.

Additional information

Original Russian Text © V.V. Fokicheva. 2014, published in Vestnik Moskovskogo Universiteta. Matematika. Mekhanika. 2014, Vol. 69, No. 4, pp. 18–27.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fokicheva, V.V. Description of singularities for billiard systems bounded by confocal ellipses or hyperbolas. Moscow Univ. Math. Bull. 69, 148–158 (2014). https://doi.org/10.3103/S0027132214040020

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0027132214040020

Keywords

Search

Navigation