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.
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Birkhoff, G.: Dynamical Systems. New York: AMS, 1927
Bunimovich, L.A.: Billiards that are close to scattering billiards. (Russian) Mat. Sb. (N.S.) 94(136), 49–73 (1974)
Deryabin, M.V., Pustyl’nikov, L.D.: Generalized relativistic billiards in external force fields. BiBoS-Preprint, No. 02-06-091, Universität Bielefeld, BiBoS, 2002
Deryabin, M.V., Pustyl’nikov, L.D.: On generalized relativistic billiards in external force fields. Lett. Math. Phys. 63(3), 195–207 (2003)
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)
Fermi, E.: On the origin of the cosmic radiation. Phys. Rev. 75, 1169–1174 (1949)
Guckenheimer, J., Holms, P.: Nonlinear Oscillations, Dynamical Systems,and Bifurcations of Vector Fields. Berlin-Heidelberg-New York: Springer-Verlag, 1983
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
Krüger, T., Pustyl’nikov, L.D., Troubetzkoy, S.E.: Acceleration of bouncing balls in external fields. Nonlinearity 8, 397–410 (1995)
Landau, L.D., Lifshitz, E.M.: The Classical theory of Fields. Oxford: Pergamon Press, 1962
Lichtenberg, A.J., Lieberman, M.A.: Regular and Chaotic Dynamics. New York: Springer, 1992
Poincaré, H.: Réflexions sur la théorie cinétique des gaz. J. Phys. Theoret. et Appl. (4) 5, 349–403 (1906)
Pustyl’nikov, L.D.: The law of entropy increase and generalized billiards. Russ. Math. Surveys 54(3), 650–651 (1999)
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)
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)
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)
Pustyl’nikov, L.D.: On a problem of Ulam. Teoret. Mat. Fiz. 57, 128–132 (1983); English transl. in Theoret. Math. Phys. 57, (1983)
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)
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)
Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Russ. Math. Surv. 25(2), 137–189 (1970)
Sinai, Ya.G., (ed.): Dynamical Systems 2. Berlin-Heidelberg-New York: Springer-Verlag, 1989, pp. 281
Sinai, Ya.G.: Introduction to ergodic theory. Princeton
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)
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)
Zaslavskii, G.M.: The stochastic property of dynamical systems. Moscow: Nauka, 1984 (Russian)
Zharnitsky, V.: Instability in Fermi-Ulam ‘ping-pong’ problem. Nonlinearity 11, 1481–1487 (1998)
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G.W. Gibbons
Dedicated to Prof. Ph. Blanchard on the occasion of his 60th birthday
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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
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DOI: https://doi.org/10.1007/s00220-004-1100-0