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Generalized relativistic billiards. (English) Zbl 1047.82021

The authors study generalized relativistic billiard, which is the following dynamical system. A particle moves in the interior of a domain under the influence of some force fields. 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. Both the motion in the domain and the reflection are considered in the framework of the theory of relativity. The authors study the periodic and ‘monotone’ action of the boundary for the particle moving in a parallelepiped and in an arbitrary compact domain respectively, and they also consider an ‘accelerating’ model in an unbounded domain. The authors prove that under some general conditions an invariant manifold in the velocity phase space of the generalized billiard, where the particle velocity equals the velocity of light, either is an exponential attractor or contains one. Thus for an open set of initial conditions the particle energy tends to infinity.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
83A05 Special relativity
83C10 Equations of motion in general relativity and gravitational theory
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
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