Abstract
In this Letter, we study generalized relativistic billiards: as a particle reflects from the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall, considered within the framework of the special theory of relativity. Inside the domain, the particle moves under the influence of some gravitational and nongravitational force fields.
We study both periodic and 'monotone' action of the boundary. We prove that under some general conditions the invariant manifold in the velocity phase space of the generalized billiard, where the point velocity equals the velocity of light, is an exponential attractor, and for an open set of initial conditions the particle energy tends to infinity.
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References
Birkhoff, G.: Dynamical Systems, Amer. Math. Soc., New York, 1927.
Bunimovich, L.A.: Billiards that are close to scattering billiards (Russian), Mat. Sb. (N. S.) 94(136)(1974), 49–73.
Deryabin, M.V. and Pustyl'nikov, L.D.: Generalized relativistic billiards in external force fields, mp-arc, 02-282.
Deryabin, M.V. and Pustyl'nikov, L.D.: Generalized relativistic billiards in external force fields. BiBoS-Preprint, No. 02-06-091, Universität Bielefeld, BiBoS, 2002.
Kozlov, V.V. and Treshchëv, D.V.: Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts, Transl. Math. Monogr., 89. Amer. Math. Soc., Providence, RI, 1991.
Krüger, T., Pustyl'nikov, L.D. and Troubetzkoy, S.E.: Acceleration of bouncing balls in external fields, Nonlinearity 8(1995), 397–410.
Landau, L.D. and Lifshitz, E.M.: The Classical Theory of Fields, Pergamon Press, Oxford, 1962.
Poincaré, A.: Réexions sur la théorie cinétique des gaz, J. Phys. Theoret. Appl. 5(4) (1906), 349–403.
Pustyl'nikov, L.D.: The law of entropy increase and generalized billiards, Russian Math. Surveys 54(3)(1999), 650–651.
Pustyl'nikov, L.D.: Poincaré models, rigorous justification of the second law of thermo-dynamics from mechanics, and the Fermi acceleration mechanism, Russian Math. Surveys 50(1)(1995), 145–189.
Pustyl'nikov, L.D.: Stable and oscillating motions in nonautonomous dynamical systems. II. (Russian), Trudy Moskov. Mat. Obšč. 34(1977), 3–103. English transl. in Trans. Moscow Math. Soc.(1978), No 2.
Pustyl'nikov, L.D.: A new mechanism for particle acceleration and a relativistic analogue of the Fermi-Ulam model, Theoret. Math. Phys. 77(1)(1988), 1110–1115.
Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards, Russ. Math. Surv. 25(2)(1970), 137–189.
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Deryabin, M.V., Pustyl'nikov, L.D. On Generalized Relativistic Billiards in External Force Fields. Letters in Mathematical Physics 63, 195–207 (2003). https://doi.org/10.1023/A:1024483416717
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DOI: https://doi.org/10.1023/A:1024483416717