A simple harmonic oscillator of mass $m$ and spring constant $k$ has the equation of motion

$$m \ddot{x}=-k x .$$

(a) Describe the orbits of the system in phase space. State how the action $I$ of the oscillator is related to a geometrical property of the orbits in phase space. Derive the action-angle variables $(\theta, I)$ and give the form of the Hamiltonian of the oscillator in action-angle variables.

(b) Suppose now that the spring constant $k$ varies in time. Under what conditions does the theory of adiabatic invariance apply? Assuming that these conditions hold, identify an adiabatic invariant and determine how the energy and amplitude of the oscillator vary with $k$ in this approximation.