The Variation of the Adiabatic Invariant of the Harmonic Oscillator

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Abstract
The variation of the adiabatic invariant μ of a harmonic oscillator or gyrating particle in a time varying homogeneous magnetic field is given as an absolutely converging series. This solution is used to discuss the behaviour of the adiabatic invariant for slow and fast as well as small and large variations of the oscillator strength. General estimates are obtained for the dependence of the oscillator strength which is twice differentiable or which is analytic. For the latter case it is found that Δμ tends to zero at least as exp [ — (2 d — ε)/α] for α → 0; 2 d is the width of the analytic strip of ω (α t); α characterizes the slowness of the variation and ε is any number 2 d > ε > 0.

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