Skip to main content
Log in

Bifurcations of Liouville tori in elliptical billiards

  • Articles
  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

A detailed description of topology of integrable billiard systems is given. For elliptical billiards and geodesic billiards on ellipsoid, the corresponding Fomenko graphs are constructed.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Arnol’d, V. I., Mathematical Methods of Classical Mechanics, Moscow: Nauka, 1989, 3rd ed. (in Russian) [New York: Springer, 1978].

    MATH  Google Scholar 

  2. Jacobi, C., Vorlesungen über Dynamic. Gesammelte Werke, Supplementband, Berlin, 1884.

  3. Cayley, A., Developments on the Porism of the In-and-circumscribed Polygon, Phil. Mag., 1854, vol. 7, pp. 339–345.

    Google Scholar 

  4. Darboux, G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitesimal, Paris: Gauthier-Villars, 1914, vols. 2 and 3.

    MATH  Google Scholar 

  5. Lebesgue, H., Les Coniques, Paris: Gauthier-Villars, 1942.

    MATH  Google Scholar 

  6. Poncelet, J.V., Traité des propriétés projectives des figures, Mett, Paris, 1822.

  7. Abenda, S. and Fedorov, Y., Closed Geodesics and Billiards on Quadrics Related to Elliptic KdV Solutions, Lett. Math. Phys., 2006, vol. 76, pp. 111–134.

    Article  MATH  MathSciNet  Google Scholar 

  8. Audin, M., Courbes algébriques et systémes intégrables: géodesiques des quadriques, Expo. Math., 1994, vol. 12, pp. 193–226.

    MATH  MathSciNet  Google Scholar 

  9. Chang, S.-J., Crespi, B., and Shi, K.-J., Elliptical Billiard Systems and the Full Poncelet’s Theorem in n Dimensions, J. Math. Phys., vol. 34, no. 6, 1993, pp. 2242–2256.

    Article  MATH  MathSciNet  Google Scholar 

  10. Chang, S.-J. and Shi, K.-J., Billiard Systems on Quadric Surfaces and the Poncelet Theorem, J. Math. Phys., 1989, vol. 30, no. 4, pp. 798–804.

    Article  MATH  MathSciNet  Google Scholar 

  11. Dragović, V. and Radnović, M., Geometry of Integrable Billiards and Pencils of Quadrics, Journal Math. Pures Appl., vol. 85, 2006, pp. 758–790.

    MATH  Google Scholar 

  12. Dragović, V. and Radnović, M., Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms, Adv. Math., 2008, vol. 219, no. 5, pp. 1577–1607.

    Article  MATH  MathSciNet  Google Scholar 

  13. Griffiths, P. and Harris, J., A Poncelet Theorem in Space, Comment. Math. Helvetici, 1977, vol. 52, no. 2, pp. 145–160.

    Article  MATH  MathSciNet  Google Scholar 

  14. Griffiths, P. and Harris, J., On Cayley’s Explicit Solution to Poncelet’s Porism, Enseign. Math., 1978, vol. 24, no. 1–2, pp. 31–40.

    MATH  MathSciNet  Google Scholar 

  15. Knörrer, H., Geodesics on the Ellipsoid, Inventiones Math., 1980, vol. 59, pp. 119–143.

    Article  MATH  Google Scholar 

  16. Moser, J., Geometry of quadrics and spectral theory, In The Chern Symposium, Springer, New York-Berlin, 1980, pp. 147–188.

    Google Scholar 

  17. Moser, J. and Veselov, A., Discrete Versions of Some Classical Integrable Systems and Factorization of Matrix Polynomials, Comm. Math. Phys., 1991, vol. 139, no. 2, pp. 217–243.

    Article  MATH  MathSciNet  Google Scholar 

  18. Previato, E., Some Integrable Billiards, in SPT2002: Symmetry and Perturbation Theory, Abenda, S., Gaeta, G., Walcher, S., Eds., Singapore: World Sci., 2002, pp. 181���195.

    Google Scholar 

  19. Waalkens, H. and Dullin, H.R., Quantum Monodromy in Prolate Ellipsoidal Billiards, Annals of Physics, 2002, vol. 295, no. 1, pp. 81–112.

    Article  MATH  MathSciNet  Google Scholar 

  20. Bolsinov, A.V. and Fomenko, A.T., Integrable Hamiltonian Systems: Geometry, Topology, Classification, Boca Roton, FL: Chapman and Hall/CRC, 2004.

    MATH  Google Scholar 

  21. Bolsinov, A.V., Matveev, S.V., and Fomenko, A.T., Topological Classification of Integrable Hamiltonian Systems with Two Degrees of Freedom. List of Systems with Small Complexity, Uspekhi Mat. Nauk, 1990, vol. 45, no. 2(272), pp. 49–77 [Russian Math. Surveys, 1990, vol. 45, no. 2, pp. 59–94].

    MATH  MathSciNet  Google Scholar 

  22. Bolsinov, A.V. and Oshemkov, A.A., Singularities of Integrable Hamiltonian Systems, in Topological Methods in the Theory of Integrable Systems, Cambridge: Camb. Sci. Publ., 2006, pp. 1–67.

    Google Scholar 

  23. Shlizerman, E. and Rom-Kedar, V., Hierarchy of Bifurcations in the Truncated and Forced Nonlinear Schrödinger Model, Chaos, 2005, vol. 15, no. 1, 013107, 22 pp.

  24. Radnović, M. and Rom-Kedar, V., Foliations of Isonergy Surfaces and Singularities of Curves, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 645–668.

    Article  MathSciNet  Google Scholar 

  25. Kozlov, V. and Treshchëv, D., Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts, Translations of Mathematical Monographs, vol. 89, Providence RI: Amer. Math. Soc., 1991.

    MATH  Google Scholar 

  26. Berger, M., Geometry, Berlin: Springer, 1987.

    Google Scholar 

  27. Bolotin, S. V., Integrable Birkhoff Billiards, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1990, no. 2, pp. 33–36 (in Russian).

  28. Turaev, D. and Rom-Kedar, V., Soft Billiards with Corners, J. Stat. Phys., 2003, vol. 112, no. 3–4, pp. 765–813.

    Article  MATH  MathSciNet  Google Scholar 

  29. Bolsinov, A.V. and Fomenko, A. T., The Geodesic Flow of an Ellipsoid is Orbitally Equivalent to the Euler Integrable Case in the Dynamics of a Rigid Body, Dokl. Akad. Nauk, 1994, vol. 339, no. 3, pp. 253–296 [Russian Acad. Sci. Dokl. Math., 1995, vol. 50, no. 3, pp. 412–417].

    MathSciNet  Google Scholar 

  30. Knörrer, H., Singular Fibres of the Momentum Mapping for Integrable Hamiltonian Systems, J. Reine Angew. Math., vol. 355, 1985, pp. 67–107.

    MATH  MathSciNet  Google Scholar 

  31. Delshams, A., Fedorov, Y., and Ramírez-Ros, R., Homoclinic Billiard Orbits Inside Symmetrically Perturbed Ellipsoids, Nonlinearity, 2001, vol. 14, pp. 1141–1195.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. Dragović.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dragović, V., Radnović, M. Bifurcations of Liouville tori in elliptical billiards. Regul. Chaot. Dyn. 14, 479–494 (2009). https://doi.org/10.1134/S1560354709040054

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354709040054

MSC2000 numbers

Key words

Navigation