(a) Show that the Hamiltonian

$$H=\frac{1}{2} p^{2}+\frac{1}{2} \omega^{2} q^{2},$$

where $\omega$ is a positive constant, describes a simple harmonic oscillator with angular frequency $\omega$. Show that the energy $E$ and the action $I$ of the oscillator are related by $E=\omega I$.

(b) Let $0<\epsilon<2$ be a constant. Verify that the differential equation

$$\ddot{x}+\frac{x}{(\epsilon t)^{2}}=0 \quad \text { subject to } \quad x(1)=0, \quad \dot{x}(1)=1$$

is solved by

$$x(t)=\frac{\sqrt{t}}{k} \sin (k \log t)$$

when $t>1$, where $k$ is a constant you should determine in terms of $\epsilon$.

(c) Show that the solution in part (b) obeys

$$\frac{1}{2} \dot{x}^{2}+\frac{1}{2} \frac{x^{2}}{(\epsilon t)^{2}}=\frac{1-\cos (2 k \log t)+2 k \sin (2 k \log t)+4 k^{2}}{8 k^{2} t}$$

Hence show that the fractional variation of the action in the limit $\epsilon \ll 1$ is $O(\epsilon)$, but that these variations do not accumulate. Comment on this behaviour in relation to the theory of adiabatic invariance.