Relativistic cyclotron orbit (${\omega }_{\tau }=qB/m$)

In a constant magnetic field $B\hat{z}$ with no electric field, a charged relativistic particle moves in a circle in the perpendicular plane. Its angular frequency with respect to proper time iswhereas its angular frequency in an inertial coordinate time is ${\omega }_t=qB/(\gamma m)$. The orbit radius at speed $v$ is

For $qB>0$, choose $A_y=Bx$ and parametrize one clockwise orbit by $x=R\cos ({\omega }_{\tau }\tau )$ and $y=-R\sin ({\omega }_{\tau }\tau )$. Over $\Delta \tau =2\pi /{\omega }_{\tau }$, the relativistic charged-particle action isThe same formula with $|{\omega }_{\tau }|$ applies for either orbit orientation.