GORELYSHEV, I. and NEISHTADT, A., 2008. Jump in adiabatic invariant at a transition between modes of motion for systems with impacts. Nonlinearity, 21(4), pp.661-676.
The one-dimensional motion of a particle between slowly moving reflecting
walls in a slowly varying potential is considered. It is shown that standard
adiabatic perturbation theory describes dynamics in the mode with impacts with
the same accuracy as in the mode without impacts. An asymptotic formula for
the jump of adiabatic invariant due to the change in mode of the motion is