Two Singular Invariants of Killing Magnetic Curves on Lightcone for Electromagnetic Design

This paper investigates the singularity properties and contact structure of spacelike Killing magnetic curves on the lightcone in Minkowski space. By employing unfolding theory, two geometric invariants are constructed to classify the singularities of magnetic trajectories. It is proved that the singularities of the principal normal magnetic surface and the revising magnetic surface are locally diffeomorphic to a cuspidal edge and a swallowtail respectively based on the vanishing conditions of geometric invariants. A concrete example is provided. The research helps predict field concentrations and structural instabilities in electromagnetic systems, and offers new tools for analyzing magnetic field interactions in relativistic settings to optimize electromagnetic device design.