Fermi–Walker magnetic curves and Killing trajectories in 3D Riemannian manifolds

Abstract

This paper deals with obtaining some new characterizations on the dynamics of the moving charged particle acting under particular forces in the three‐dimensional space. In this perspective, we have enlarged the research of magnetic curves and associated Killing magnetic fields by adding the Fermi–Walker parallelism and derivative. By doing this, we investigate the Hall–Killing–Fermi effect on the charged particle, then we define new types of Fermi–Walker magnetic curves corresponding to the Killing magnetic field. We carefully construct the Fermi–Walker character of the magnetic field to prove the parallelism of the magnetic trajectories and uniformly accelerated motion of the charged particle. It leads to defining a very special family of magnetic fields, that is, Fermi–Walker uniform magnetic fields. Moreover, we use the Killing feature of the magnetic field to show that the Frenet–Serret scalars and quasislopes of the magnetic curves must meet particular differential equations that lead to obtaining some geometric interpretations of the magnetic trajectories. Last but not the least, both of these two characters allow us to define Killing and Fermi–Walker electric vector fields influencing the motion of the charged particle experiencing the Lorentz force in the three‐dimensional setting.