Adiabatic approximation for a model of cyclotron motion

Aspecific problem is used to illustrate the limits of the approach resulting in an adiabatic approximation. The system of differential equations modeling the cyclotron motion of a charged relativistic particle in the field of an electromagnetic wave is considered. The problem of resonance capture of a particle with significantly varying energy is studied. The main result is the description of the capture area, i.e., the set of initial points in the phase plane from which the resonance trajectories issue. Such a description is obtained by the method of asymptotic approximation in a small parameter which corresponds to the rate of variation in the magnetic field. It is discovered that such an approximation is inapplicable in the case of small amplitudes of the electromagnetic wave.