Motion of charged particles in homogeneous electromagnetic fields

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Abstract
An invariant geometrical description of the world lines of charged particles in arbitrary homogeneous electromagnetic fields is presented. This is accomplished through the combined use of the Frenet‐Serret equations and the Lorentz equation. The results apply to flat as well as Riemannian space‐time. The intrinsic scalars associated with these curves (i.e., their curvatures and first and second torsions) are found to be constants of the motion when they are well defined. Moreover, they form simple relationships with the field invariants as well as with the energy and momentum densities of the rest frame fields. When they are evaluated in the instantaneous rest frame of the particle, the Frenet vectors lend themselves to simple physical interpretation. It is shown that one cannot distinguish in an intrinsic geometrical manner between the curves of positive and negative charges. The same is true for positive and negative magnetic monopoles if they exist. In such a case, however, one would be able to distinguish intrinsically between ordinary and magnetic charges. The effect of duality rotations of the field tensor on the Frenet scalars is studied. A physical realization of the Frenet frame is obtained by considering the classical description of spin precession. Finally the Frenet formalism is applied to timelike Killing trajectories. These are shown to closely resemble the world lines of charged particles in homogeneous electromagnetic fields.

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