Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 14D / c / Solution

Put $\theta =k\log t$. Sincethe instantaneous oscillator energy isThe instantaneous frequency is $\omega (t)=1/(ϵt)$, soIts oscillatory part divided by its mean is $O(k^{-1})=O(ϵ)$. It remains bounded and periodic in $\log t$, so the variations do not accumulate. This is the expected behavior of an adiabatic invariant: slow parameter variation produces small bounded oscillations rather than secular drift.