  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.
