An extended adiabatic invariant

The motion of a one dimensional oscillator with time dependent restoring force is investigated. We consider ω( t ) having one singularity for finite t and such that lim | t |-> ∞ | d / dt ω -1 ( t )| = 0 and consider the asymptotic behaviour of ω times the square of the amplitude. The ratio of this quantity evaluated at two times, t 1 and t 2 averaged over an ensemble is, in the limit | t 1 |, | t 2 | → ∞ , an adiabatic invariant in the sense that this quantity is not changed by multiplication of ω by a slowly varying function of t . This permits replacement of the actual ω( t ) by a, presumably simpler, ω 0 ( t ) if their ratio, ω 0 /ω is non-singular. Since the orbit equation for a charge in a time-dependent axisymmetric magnetic field is reducible to that of a one-dimensional oscillator it possesses such an invariant. This problem is examined in some detail, particularly for the model field ω( t ) = ω t . These solutions provide explicit estimates for the energy gained by a charge under a reversal of the magnetic field.