A particle of rest mass $m$ and charge $q$ moves along a path $x^{a}(s)$, where $s$ is the particle's proper time. The equation of motion is

$$m \ddot{x}^{a}=q F^{a b} \eta_{b c} \dot{x}^{c}$$

where $\dot{x}^{a}=d x^{a} / d s$ etc., $F^{a b}$ is the Maxwell field tensor $\left(F^{01}=-E_{x}, F^{23}=-B_{x}\right.$, where $E_{x}$ and $B_{x}$ are the $x$-components of the electric and magnetic fields) and $\eta_{b c}$ is the Minkowski metric tensor. Show that $\dot{x}_{a} \ddot{x}^{a}=0$ and interpret both the equation of motion and this equation in the classical limit.

The electromagnetic field is given in cartesian coordinates by $\mathbf{E}=(0, E, 0)$ and $\mathbf{B}=(0,0, E)$, where $E$ is constant and uniform. The particle starts from rest at the origin. Show that the orbit is given by

$$9 x^{2}=2 \alpha y^{3}, \quad z=0$$

where $\alpha=q E / m$.