Equations of motion of a relativistic charged particle with a curvature depending actions

Abstract

We present an introduction to the study of a relativistic particle moving under the influence of its own Frenet-Serret curvatures. With the aim of introducing the notation and conventions used in this paper, we first recall the action of a relativistic particle. We then suggest a mathematical generalization of this action in the sense that now the action may include terms of the curvatures of the world line generated by the particle in Minkowski space-time. We go on to develop a pedagogical introduction to a variational calculus which takes advantage of the Frenet-Serret equations for the relativistic particle. Finally, we consider a relativistic particle coupled to an electromagnetic field that is moving under the influence of its own Frenet-Serret curvatures. Within this frame based on the Frenet-Serret basis, we obtain the equations of motion for several curvature dependent actions of interest in physics. Later, as an illustration of the formalism developed, we consider the simplest case, that of a relativistic particle when no geometrical action is included, in order to show (i) the equivalence of this formalism to the Newton's second law with the Lorentz force and (ii) the integrability in the case of a constant electromagnetic field.