(a) Consider a system with one degree of freedom, which undergoes periodic motion in the potential $V(q)$. The system's Hamiltonian is

$$H(p, q)=\frac{p^{2}}{2 m}+V(q)$$

(i) Explain what is meant by the angle and action variables, $\theta$ and $I$, of the system and write down the integral expression for the action variable $I$. Is $I$ conserved? Is $\theta$ conserved?

(ii) Consider $V(q)=\lambda q^{6}$, where $\lambda$ is a positive constant. Find $I$ in terms of $\lambda$, the total energy $E$, the mass $M$, and a dimensionless constant factor (which you need not compute explicitly).

(iii) Hence describe how $E$ changes with $\lambda$ if $\lambda$ varies slowly with time. Justify your answer.

(b) Consider now a particle which moves in a plane subject to a central force-field $\mathbf{F}=-k r^{-2} \hat{\mathbf{r}}$.

(i) Working in plane polar coordinates $(r, \phi)$, write down the Hamiltonian of the system. Hence deduce two conserved quantities. Prove that the system is integrable and state the number of action variables.

(ii) For a particle which moves on an elliptic orbit find the action variables associated with radial and tangential motions. Can the relationship between the frequencies of the two motions be deduced from this result? Justify your answer.

(iii) Describe how $E$ changes with $m$ and $k$ if one or both of them vary slowly with time.

[You may use

$$\int_{r_{1}}^{r_{2}}\left\{\left(1-\frac{r_{1}}{r}\right)\left(\frac{r_{2}}{r}-1\right)\right\}^{\frac{1}{2}} d r=\frac{\pi}{2}\left(r_{1}+r_{2}\right)-\pi \sqrt{r_{1} r_{2}}$$

where $0<r_{1}<r_{2}$.]