Write down the equation of motion for a point particle with mass $m$, charge $e$, and position vector $\mathbf{x}(t)$ moving in a time-dependent magnetic field $\mathbf{B}(\mathbf{x}, t)$ with vanishing electric field, and show that the kinetic energy of the particle is constant. If the magnetic field is constant in direction, show that the component of velocity in the direction of $\mathbf{B}$ is constant. Show that, in general, the angular momentum of the particle is not conserved.

Suppose that the magnetic field is independent of time and space and takes the form $\mathbf{B}=(0,0, B)$ and that $\dot{A}$ is the rate of change of area swept out by a radius vector joining the origin to the projection of the particle's path on the $(x, y)$ plane. Obtain the equation

$$\frac{d}{d t}\left(m \dot{A}+\frac{e B r^{2}}{4}\right)=0,$$

where $(r, \theta)$ are plane polar coordinates. Hence obtain an equation replacing the equation of conservation of angular momentum.

Show further, using energy conservation and $(*)$, that the equations of motion in plane polar coordinates may be reduced to the first order non-linear system

$$\begin{gathered} \dot{r}=\sqrt{v^{2}-\left(\frac{2 c}{m r}-\frac{e r B}{2 m}\right)^{2}} \\ \dot{\theta}=\frac{2 c}{m r^{2}}-\frac{e B}{2 m} \end{gathered}$$

where $v$ and $c$ are constants.