= Solution
For $A^\mu=(0,0,Bx,0)$, the only independent nonzero component of the <electromagnetic field tensor> is
$$
F^{12}=B,
\qquad F^{21}=-B.
$$
The temporal and $z$ components of the <Lorentz force> show that $u^0$ and $u^3$ are constant. Taking $u^3=0$, the transverse equations are
$$
\ddot x=\frac{qB}{m}\dot y,
\qquad
\ddot y=-\frac{qB}{m}\dot x,
$$
where dots now denote <proper time> derivatives. With
$$
\boxed{\omega=\frac{qB}{m}},
$$
a solution is
$$
\boxed{
x(\tau)=x_c+R\cos\bigl(\omega(\tau-\tau_0)\bigr),
\qquad
y(\tau)=y_c-R\sin\bigl(\omega(\tau-\tau_0)\bigr).
}
$$
Direct differentiation verifies both equations. Hence the trajectory is a <relativistic cyclotron orbit> with proper angular frequency $omega=qB/m$; the sign fixes its orientation.
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