Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 8E (Classical Dynamics)

LetAn orbit of energy $E>0$ is an ellipse in phase space, with intercepts $x=\pm \sqrt{2E/(m{\omega }^2)}$ and $p=\pm \sqrt{2mE}$. Its enclosed area is $2\pi E/\omega$, so the action variable isThe transformationis canonical and gives the action-angle variables. Substitution yields

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 $k^{-1/4}$.