Adiabatic particle orbits in discontinuous fields

The invariance properties of the action integral J = $\oint$ p dq are studied for the motion of charged particles in one-dimensional electromagnetic fields. Attention is concentrated on those situations where the field gradients become large. Whereas in the case of smooth fields and finite gradients the asymptotic theory of the one-dimensional oscillator as developed by Gardner and by Lenardapplies, the presence of large gradients, requires a special treatment. The present considerations are restricted to cases where the strong field variation is confined to a small region such that the transition n can be approximated by a discontinuity. lt is shown that for time intervals of order I/$\epsilon$, J is an adiabatic invariant of at least order $\epsilon^{\frac{1}{2}}$, $\epsilon$ measuring the time variation of the fields. As an example, the motion of high-speed particles through a plane discontinuity is shown to be adiabatic in that sense.Consequently, the initial and the final magnetic moments differ only slightly