A particle of rest mass $m$ and charge $q$ moves in an electromagnetic field given by a potential $A_{a}$ along a trajectory $x^{a}(\tau)$, where $\tau$ is the proper time along the particle's worldline. The action for such a particle is

$$I=\int\left(m \sqrt{-\eta_{a b} \dot{x}^{a} \dot{x}^{b}}-q A_{a} \dot{x}^{a}\right) d \tau .$$

Show that the Euler-Lagrange equations resulting from this action reproduce the relativistic equation of motion for the particle.

Suppose that the particle is moving in the electrostatic field of a fixed point charge $Q$ with radial electric field $E_{r}$ given by

$$E_{r}=\frac{Q}{4 \pi \epsilon_{0} r^{2}} .$$

Show that one can choose a gauge such that $A_{i}=0$ and only $A_{0} \neq 0$. Find $A_{0}$.

Assume that the particle executes planar motion, which in spherical polar coordinates $(r, \theta, \phi)$ can be taken to be in the plane $\theta=\pi / 2$. Derive the equations of motion for $t$ and $\phi$.

By using the fact that $\eta_{a b} \dot{x}^{a} \dot{x}^{b}=-1$, find the equation of motion for $r$, and hence show that the shape of the orbit is described by

$$\frac{d r}{d \phi}=\pm \frac{r^{2}}{\ell} \sqrt{\left(E+\frac{\gamma}{r}\right)^{2}-1-\frac{\ell^{2}}{r^{2}}}$$

where $E(>1)$ and $\ell$ are constants of integration and $\gamma$ is to be determined.

By putting $u=1 / r$ or otherwise, show that if $\gamma^{2}<\ell^{2}$ then the orbits are bounded and generally not closed, and show that the angle between successive minimal values of $r$ is $2 \pi\left(1-\gamma^{2} / \ell^{2}\right)^{-1 / 2}$.