An Overview of the Behaviour of a Scattering Map for the Dynamics of Two Interacting Particles in a Uniform Magnetic Field

Abstract

The interaction of two charges moving in ℝ3 in a magnetic field B can be formulated as a Hamiltonian system with six degrees of freedom. A scattering map is defined for trajectories which come from and go to infinite separation along the field direction. It determines the asymptotic parallel velocities, guiding centre field lines, magnetic moments and gyrophases for large positive time from those for large negative time. In regimes where gyrophase averaging is appropriate, the scattering map has a simple form, conserving the magnetic moments and parallel kinetic energies (in a frame moving along the field with the centre of mass) and rotating or translating the guiding centre field lines. When the gyrofrequencies are in low order resonance, however, gyrophase averaging is not justified and transfer of perpendicular kinetic energy occurs. In the extreme case of equal gyrofrequencies there is typically also transfer between perpendicular and parallel kinetic energy.