Maxwell-Lorentz equations in general Frenet-Serret coordinates

We consider the trajectory of a charged particle in an arbitrary external magnetic field. A local orthogonal coordinate system is given by the tangential, curvature, and torsion vectors. We write down Maxwell´s equations in this coordinate system. The resulting partial differential equations for the magnetic fields fix conditions among its local multipole components, which can be viewed as a generalization of the usual multipole expansion of the fields of magnetic elements.