A particle of mass $m$, charge $e$ and position vector $\mathbf{r}=\left(x_{1}, x_{2}, x_{3}\right) \equiv \mathbf{q}$ moves in a magnetic field whose vector potential is A. Its Hamiltonian is given by

$$H(\mathbf{p}, \mathbf{q})=\frac{1}{2 m}\left(\mathbf{p}-e \frac{\mathbf{A}}{c}\right)^{2}$$

Write down Hamilton's equations and use them to derive the equations of motion for the charged particle.

Define the Poisson bracket $[F, G]$ for general $F(\mathbf{p}, \mathbf{q})$ and $G(\mathbf{p}, \mathbf{q})$. Show that for motion governed by the above Hamiltonian

$$\left[m \dot{x}_{i}, x_{j}\right]=-\delta_{i j}, \quad \text { and } \quad\left[m \dot{x}_{i}, m \dot{x}_{j}\right]=\frac{e}{c}\left(\frac{\partial A_{j}}{\partial x_{i}}-\frac{\partial A_{i}}{\partial x_{j}}\right)$$

Consider the vector potential to be given by $\mathbf{A}=(0,0, F(r))$, where $r=\sqrt{x_{1}^{2}+x_{2}^{2}}$. Use Hamilton's equations to show that $p_{3}$ is constant and that circular motion at radius $r$ with angular frequency $\Omega$ is possible provided that

$$\Omega^{2}=-\left(p_{3}-\frac{e F}{c}\right) \frac{e}{m^{2} c r} \frac{d F}{d r}$$