The motion of a particle of charge $q$ and mass $m$ in an electromagnetic field with scalar potential $\phi(\mathbf{r}, t)$ and vector potential $\mathbf{A}(\mathbf{r}, t)$ is characterized by the Lagrangian

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

(i) Write down the Hamiltonian of the particle.

(ii) Write down Hamilton's equations of motion for the particle.

(iii) Show that Hamilton's equations are invariant under the gauge transformation

$$\phi \rightarrow \phi-\frac{\partial \Lambda}{\partial t}, \quad \mathbf{A} \rightarrow \mathbf{A}+\nabla \Lambda,$$

for an arbitrary function $\Lambda(\mathbf{r}, t)$.

(iv) The particle moves in the presence of a field such that $\phi=0$ and $\mathbf{A}=\left(-\frac{1}{2} y B, \frac{1}{2} x B, 0\right)$, where $(x, y, z)$ are Cartesian coordinates and $B$ is a constant.

(a) Find a gauge transformation such that only one component of $\mathbf{A}(x, y, z)$ remains non-zero.

(b) Determine the motion of the particle.

(v) Now assume that $B$ varies very slowly with time on a time-scale much longer than $(q B / m)^{-1}$. Find the quantity which remains approximately constant throughout the motion.

[You may use the expression for the action variable $I=\frac{1}{2 \pi} \oint p_{i} d q_{i}$.]