The study of charged-particle motion in electromagnetic fields is a rich source of problems, models, and new phenomena for nonlinear dynamics. The case of a strong magnetic field is well studied in the framework of a guiding center theory, which is based on conservation of an adiabatic invariant—the magnetic moment. This theory ceases to work near a line on which the magnetic field vanishes—the magnetic field null line. In this paper, we show that the existence of these lines leads to remarkable phenomena which are new both for nonlinear dynamics in general and for the theory of charged-particle motion. We consider the planar motion of a charged particle in a strong stationary perpendicular magnetic field with a null line and a strong electric field. We show that particle dynamics switch between a slow guiding center motion and the fast traverse along a segment of the magnetic field null line. This segment is the same (in the principal approximation) for all particles with the same total energy. During the phase of a guiding center motion, the magnetic moment of particle’s Larmor rotation stays approximately constant, i.e., it is an adiabatic invariant. However, upon each traversing of the null line, the magnetic moment changes in a random fashion, causing the particle to choose a new trajectory of the guiding center motion. This results in a stationary distribution of the magnetic moment, which only depends on the particle’s total energy. The jumps in the adiabatic invariant are described by Painlevé II equation.
The existence of adiabatic invariants—approximate conservation laws for systems with slow and fast motions—plays an important role in different physical theories. One of such theories is guiding center theory of motion of charged particles in a strong magnetic field. This theory is based on the adiabatic invariance of magnetic moment for the particle motion. Basic assumption of the guiding center approach is that the magnetic field is strong and vanishes nowhere. We show that, for a planar motion in strong perpendicular magnetic fields, if the magnetic field vanishes on some line (magnetic field null line) and the strong electric field is present, then the particle gets involved in a peculiar process of capture and release by the null line, which leads to large chaotic oscillations of the particle magnetic moment. Such a behaviour has not been previously reported in charged particles dynamics or in nonlinear dynamics in general.