NONLINEAR-THEORY OF PARTICLE DYNAMICS IN COSMIC ELECTROMAGNETIC-FIELDS .1. REGULAR MOTION IN A SIMPLE ALFVEN-WAVE

With this paper, we begin a study of the non-linear dynamics of the non-thermal component of the cosmic environment. The problem of a test particle propagating in a sinusoidal Alfven wave is of basic importance in the perturbation theory of cosmic ray dynamics. In fact, it is assumed that the character of the motion is diffusive because of resonant pitch angle scattering off fluctuations of the cosmic magnetic field. The non-linear theory of this problem shows a richer behaviour. We find an approximate invariant of the motion, which accurately describes the dynamics in the vicinity of resonances. This invariant shows that particles can be locked in a resonant configuration, instead of being scattered. Only one resonance is admitted between a particle and an individual harmonic of the magnetic fluctuations, by the well known cyclotron resonance condition of quasi-linear theory. We find a non-linear resonance condition yielding an infinite number of resonances in the motion of a particle in a purely sinusoidal wave, parallel to the mean magnetic field. Our results are not restricted to the case of small amplitude perturbations.