On adiabatic perturbation theory for systems with elastic collisions

It is a remark by V.I. Arnold that, as a general phenomenon, it is more convenient to think about mappings, but easier to calculate with flows ([1], p. 394). In line with this remark, we present a procedure of adiabatic perturbation theory for systems with elastic collisions which operates with Hamilton's functions rather than with maps (e.g. billiard maps). This simplifies the exposition and makes it identical to that for smooth systems. The accuracy of the procedure is the same as for smooth systems. We consider three models where there is one fast angle variable (from a pair “action-angle” variables): (a) an one-dimensional motion of a particle between slowly moving walls (Fermi–Ulam model [2], Sect. 3.4) in a slowly varying potential field between these walls, (b) propagation of light rays in a slowly irregular planar refractional light guide with reflecting walls [3], and (c) motion of a charged particle in a planar magnetic billiard [4] with a strong non-uniform magnetic field. The key observation is that while considered Hamilton's functions have discontinuities with respect to the fast angle variable, there are no derivatives with respect to it in the procedure. The action variable is an approximate first integral of dynamics – an adiabatic invariant.
The adiabatic perturbation procedure is applicable provided that there is no change of regime of fast motion in the process of slow evolution. Assume now that there is a change of regime of motion from a regime with collisions with a wall to a regime without collisions. This is an analog of crossing of a separatrix in smooth systems. In each of regimes the system has an “improved” adiabatic invariant which is conserved with a high accuracy. Change of regime leads to a quasi-random jump in value of this adiabatic invariant. We present an asymptotic formula for this jump. Accumulation of results of many such jumps in case of multiple changes of regimes of motion in model (c) leads to destruction of adiabatic invariance. This is one of mechanisms of the origin of chaotic dynamics for systems with collisions.
The talk is a review based on joint works with I. V. Gorelyshev and A. V. Artemyev [5, 6, 7, 8].