Contribution to null Killing magnetic trajectories

Abstract

We analyzed null magnetic trajectories of a magnetic field on a timelike surface in Minkowski 3-space. We show that the Lorentz force can be written into the Darboux frame field of a null trajectory on the surface. We give the necessary and sufficient condition for writing a null curve as the magnetic trajectory of the magnetic field. Then constructing a variation, we derive the Killing magnetic flow equations with regard to the geodesic curvature, geodesic torsion and normal curvature of the curve γ on the timelike surface. Finally we examine magnetic curves on a timelike pseudo-sphere and the pseudo-cylinder.