Charged-particle action over one relativistic cyclotron orbit

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.