Invariants in the Motion of a Charged Particle in a Spatially Modulated Magnetic Field

Abstract

In this paper we study the effect of a spatial modulation in the magnetic field on invariants such as the magnetic moment ${\mathrm{\mu =V}}_{\perp }^2\mathrm{/B}$ . In particular, we investigate whether an invariant still exists when the wavelength of the modulation is comparable to the gyro radius of the particle. In an axially symmetric magnetic field, with a square‐wave modulation, the orbit equations reduce to algebraic relations convenient for numerical study. We find from such studies that orbits are of two types: (a) regular orbits which generate an invariant; (b) orbits which are quasi‐ergodic. We have also calculated an invariant by perturbation theory with the depth of modulation of the field as a small parameter. For this we developed a modified form of perturbation theory which overcomes the difficulty of infinities arising at resonance between the perturbation and the cyclotron period. This difficulty in fact corresponds to a change in topology of the invariant curves. The invariant calculated from this theory shows very good agreement with the numerically computed orbits of type (a). The transition to quasi‐ergodic behavior cannot be predicted analytically, but some indication of it may exist in the complex topology of the invariant curves in the ergodic regions.