Motion of electrons in magnetic bottle (4)

The motion of a charged particle injected in an axially symmetric magnetic bottle is investigated. The canonical formalism developed in a foregoing paper is applied to the case of an off-axis region, where the ratios of the gyroradius to the radius of curvature of the magnetic line of force and to the radial distance from the axis are considered to be small. An expansion of the Hamiltonian with respect to these ratios is performed, and use of the perturbation theory is made. The zeroth order theory is readily soluble and coincides with the usual Alfven theory of the guiding center. The first order corrections are calculated in Born-Oppenheimer's approximation, and the change in the magnetic moment associated with the gyration is given explicitly. The result obtained is in good agreement with Kruskal's general consideration. The mirror points are not fixed but are shifted at every turn according with the gyration phase. Some remarks on the higher order corrections are also given. (author)