Destruction of adiabatic invariance at resonances in slow–fast Hamiltonian systems

There are many problems that lead to analysis of dynamical systems in which one can distinguish motions of two types: slow one and fast one. An averaging over fast motion is used for approximate description of the slow motion. First integrals of the averaged system are approximate first integrals of the exact system, i.e. adiabatic invariants. Resonant phenomena in fast motion (capture into resonance, scattering on resonance) lead to inapplicability of averaging, destruction of adiabatic invariance, dynamical chaos and transport in large domains in the phase space. In the paper perturbation theory methods for description of these phenomena are outlined. We also consider as an example the problem of surfatron acceleration of a relativistic charged particle.