Adiabatic Invariant of the Harmonic Oscillator
Russell M. Kulsrud · Physical Review · 1957
The problem of a vibrating harmonic oscillator whose frequency is changing in time is considered in the case where the frequency $\ensuremath{\omega}$ is initially constant, varies in an arbitrary fashion and becomes constant again. It is found that the relative change of the quantity, the energy divided by the frequency, in the final region from its value in the initial region is zero to as many orders in the rate of change of $\ensuremath{\omega}$ as $\ensuremath{\omega}$ has continuous derivatives. For the case where there is a break in the $N\mathrm{th}$ derivative of $\ensuremath{\omega}$ the relative change is given to this order.