Skip to main content
SpringerLink
Log in

Exponential Attractors in Generalized Relativistic Billiards

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

A generalized relativistic billiard is the following dynamical system: a particle moves under the influence of some force fields in the interior of a domain with pseudo-Riemannian metric, and as the particle hits the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall, considered in the framework of the special theory of relativity. We study a periodic and a ‘‘monotone’’ action of the boundary. We prove that in both cases under some general conditions the invariant manifold in the velocity phase space of the generalized relativistic billiard, where the point velocity equals the velocity of light, is an exponential attractor or contains one, and for an open set of initial conditions the particle energy tends to infinity.

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. Birkhoff, G.: Dynamical Systems. New York: AMS, 1927

  2. Bunimovich, L.A.: Billiards that are close to scattering billiards. (Russian) Mat. Sb. (N.S.) 94(136), 49–73 (1974)

    MATH  Google Scholar 

  3. Deryabin, M.V., Pustyl’nikov, L.D.: Generalized relativistic billiards in external force fields. BiBoS-Preprint, No. 02-06-091, Universität Bielefeld, BiBoS, 2002

  4. Deryabin, M.V., Pustyl’nikov, L.D.: On generalized relativistic billiards in external force fields. Lett. Math. Phys. 63(3), 195–207 (2003)

    Article  MATH  Google Scholar 

  5. Dovbysh, S.A.: Kolmogorov stability, the impossibility of Fermi acceleration, and the existence of periodic solutions on some systems of Hamilton type. Prikl. Mat. Mekh. 56, 218–229 (1992); English transl. in J. Appl. Math. Mech. 56, (1992)

    MathSciNet  MATH  Google Scholar 

  6. Fermi, E.: On the origin of the cosmic radiation. Phys. Rev. 75, 1169–1174 (1949)

    Article  MATH  Google Scholar 

  7. Guckenheimer, J., Holms, P.: Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields. Berlin-Heidelberg-New York: Springer-Verlag, 1983

  8. Kozlov, V.V., Treshchëv, D.V.: Billiards. A genetic introduction to the dynamics of systems with impacts. Trans. Math. Monographs 89. Providence, RI: American Mathematical Society, 1991, viii+171 pp

  9. Krüger, T., Pustyl’nikov, L.D., Troubetzkoy, S.E.: Acceleration of bouncing balls in external fields. Nonlinearity 8, 397–410 (1995)

    Article  MathSciNet  Google Scholar 

  10. Landau, L.D., Lifshitz, E.M.: The Classical theory of Fields. Oxford: Pergamon Press, 1962

  11. Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. New York: Springer, 1992

  12. Poincaré, H.: Réflexions sur la théorie cinétique des gaz. J. Phys. Theoret. et Appl. (4) 5, 349–403 (1906)

    Google Scholar 

  13. Pustyl’nikov, L.D.: The law of entropy increase and generalized billiards. Russ. Math. Surveys 54(3), 650–651 (1999)

    Article  MATH  Google Scholar 

  14. Pustyl’nikov, L.D.: Poincaré models, rigorous justification of the second law of thermodynamics from mechanics, and the Fermi acceleration mechanism. Russ. Math. Surveys 50(1), 145–189 (1995)

    MATH  Google Scholar 

  15. Pustyl’nikov, L.D.: Stable and oscillating motions in nonautonomous dynamical systems. II. (Russian) Trudy Moskov. Mat. Obšč. 34, 3–103 (1977); English transl. in Trans. Moscow Math. Soc. (2), (1978)

    MATH  Google Scholar 

  16. Pustyl’nikov, L.D.: A new mechanism for particle acceleration and a relativistic analogue of the Fermi-Ulam model. Theoret. Math. Phys. 77(1), 1110–1115 (1988)

    Google Scholar 

  17. Pustyl’nikov, L.D.: On a problem of Ulam. Teoret. Mat. Fiz. 57, 128–132 (1983); English transl. in Theoret. Math. Phys. 57, (1983)

    MathSciNet  Google Scholar 

  18. Pustyl’nikov, L.D.: On the Fermi-Ulam model. Dokl. Akad. Nauk SSSR 292, 549–553 (1987); English transl. in Soviet Math. Dokl. 35, (1987)

    MathSciNet  MATH  Google Scholar 

  19. Pustyl’nikov, L.D.: The existence of invariant curves for mapping close to degenerate ones, and the solution of the Fermi-Ulam problem. Mat. Sb. 185(6), 1–12 (1994); English transl. in Russ Acad. Sci. Sb. Math. 82, (1995)

    MATH  Google Scholar 

  20. Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25(2), 137–189 (1970)

    Google Scholar 

  21. Sinai, Ya.G., (ed.): Dynamical Systems 2. Berlin-Heidelberg-New York: Springer-Verlag, 1989, pp. 281

  22. Sinai, Ya.G.: Introduction to ergodic theory. Princeton

  23. Ulam, S.M.: On some statistical properties of dynamical systems. In: Proc. 4th Berkeley Sympos.on Math.Statist.and Prob., Vol. III, Berkeley, CA: Univ. California Press, pp. 315–320 (1961)

  24. Zaslavskii, G.M., Chirikov, B.V.: The Fermi acceleration mechanism in the one-dimensional case. Dokl. Akad. Nauk SSSR 159, 306–309 (1964); English transl. in Soviet Phys. Dokl. 9, (1964)

    Google Scholar 

  25. Zaslavskii, G.M.: The stochastic property of dynamical systems. Moscow: Nauka, 1984 (Russian)

  26. Zharnitsky, V.: Instability in Fermi-Ulam ‘ping-pong’ problem. Nonlinearity 11, 1481–1487 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, 119899, Moscow, Russia

    M.V. Deryabin

  2. Keldysh Institute of Applied Mathematics of RAS, Miusskaja sq. 4, 125047, Moscow, Russia

    L.D. Pustyl’nikov

  3. University of Bielefeld, BiBoS, Postfach 100131, 33501, Bielefeld, Germany

    L.D. Pustyl’nikov

Authors
  1. M.V. Deryabin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L.D. Pustyl’nikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

G.W. Gibbons

Dedicated to Prof. Ph. Blanchard on the occasion of his 60th birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deryabin, M., Pustyl’nikov, L. Exponential Attractors in Generalized Relativistic Billiards. Commun. Math. Phys. 248, 527–552 (2004). https://doi.org/10.1007/s00220-004-1100-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-004-1100-0

Keywords

Search

Navigation