(i) The action $S$ for a point particle of rest mass $m$ and charge $q$ moving along a trajectory $x^{\mu}(\lambda)$ in the presence of an electromagnetic 4 -vector potential $A^{\mu}$ is

$$S=-m c \int\left(-\eta_{\mu \nu} \frac{d x^{\mu}}{d \lambda} \frac{d x^{\nu}}{d \lambda}\right)^{1 / 2} d \lambda+q \int A_{\mu} \frac{d x^{\mu}}{d \lambda} d \lambda$$

where $\lambda$ is an arbitrary parametrization of the path and $\eta_{\mu \nu}$ is the Minkowski metric. By varying the action with respect to $x^{\mu}(\lambda)$, derive the equation of motion $m \ddot{x}^{\mu}=q F^{\mu} \dot{x}^{\nu}$, where $F_{\mu \nu}=\partial_{\mu} A_{\nu}-\partial_{\nu} A_{\mu}$ and overdots denote differentiation with respect to proper time for the particle.

(ii) The particle moves in constant electric and magnetic fields with non-zero Cartesian components $E_{z}=E$ and $B_{y}=B$, with $B>E / c>0$ in some inertial frame. Verify that a suitable 4-vector potential has components

$$A^{\mu}=(0,0,0,-B x-E t)$$

in that frame.

Find the equations of motion for $x, y, z$ and $t$ in terms of proper time $\tau$. For the case of a particle that starts at rest at the spacetime origin at $\tau=0$, show that

$$\ddot{z}+\frac{q^{2}}{m^{2}}\left(B^{2}-\frac{E^{2}}{c^{2}}\right) z=\frac{q E}{m} .$$

Find the trajectory $x^{\mu}(\tau)$ and sketch its projection onto the $(x, z)$ plane.