A particle of rest-mass $m$, electric charge $q$, is moving relativistically along the path $x^{\mu}(s)$ where $s$ parametrises the path.

Write down an action for which the extremum determines the particle's equation of motion in an electromagnetic field given by the potential $A^{\mu}(x)$.

Use your action to derive the particle's equation of motion in a form where $s$ is the proper time.

Suppose that the electric and magnetic fields are given by

$$\begin{aligned} \mathbf{E} &=(0,0, E) \\ \mathbf{B} &=(0, B, 0) \end{aligned}$$

where $E$ and $B$ are constants and $B>E>0$.

Find $x^{\mu}(s)$ given that the particle starts at rest at the origin when $s=0$.

Describe qualitatively the motion of the particle.