(i) Consider a particle of charge $q$ and mass $m$, moving in a stationary magnetic field B. Show that Lagrange's equations applied to the Lagrangian

$$L=\frac{1}{2} m \dot{\mathbf{r}}^{2}+q \dot{\mathbf{r}} \cdot \mathbf{A}(\mathbf{r})$$

where $\mathbf{A}$ is the vector potential such that $\mathbf{B}=\operatorname{curl} \mathbf{A}$, lead to the correct Lorentz force law. Compute the canonical momentum $\mathbf{p}$, and show that the Hamiltonian is $H=\frac{1}{2} m \dot{\mathbf{r}}^{2}$.

(ii) Expressing the velocity components $\dot{r}_{i}$ in terms of the canonical momenta and co-ordinates for the above system, derive the following formulae for Poisson brackets: (b) $\left\{m \dot{r}_{i}, m \dot{r}_{j}\right\}=q \epsilon_{i j k} B_{k}$; (c) $\left\{m \dot{r}_{i}, r_{j}\right\}=-\delta_{i j}$; (d) $\left\{m \dot{r}_{i}, f\left(r_{j}\right)\right\}=-\frac{\partial}{\partial r_{i}} f\left(r_{j}\right)$.

(a) $\{F G, H\}=F\{G, H\}+\{F, H\} G$, for any functions $F, G, H$;

Now consider a particle moving in the field of a magnetic monopole,

$$B_{i}=g \frac{r_{i}}{r^{3}} .$$

Show that $\{H, \mathbf{J}\}=0$, where $\mathbf{J}=m \mathbf{r} \wedge \dot{\mathbf{r}}-g q \hat{\mathbf{r}}$. Explain why this means that $\mathbf{J}$ is conserved.

Show that, if $g=0$, conservation of $\mathbf{J}$ implies that the particle moves in a plane perpendicular to $\mathbf{J}$. What type of surface does the particle move on if $g \neq 0$ ?