Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 8E / b / Solution

The adiabatic invariance of the action applies when the frequency changes by only a small fraction during one oscillation and no separatrix is encountered. A convenient condition isThenConsequentlyIf $A$ is the displacement amplitude, $E=\frac{1}{2}k{A}^2$, and henceThus a slowly stiffened spring raises the energy as $k^{1/2}$ while reducing the amplitude as $k^{-1/4}$.