  We describe how to construct the dynamics of relativistic particles
following, either timelike or null curves, by means of an auxiliary variables
method instead of the standard theory of deformations for curves. There are
interesting physical particle models governed by actions that involve higher
order derivatives of the embedding functions of the worldline. We point out
that the mechanical content of such models can be extracted wisely from a lower
order action, which can be performed by implementing in the action a finite
number of constraints that involve the geometrical relationship structures
inherent to a curve and by using a covariant formalism. We emphasize our
approach for null curves. For such systems, the natural time parameter is a
pseudo-arclength whose properties resemble those of the standard proper time.
We illustrate the formalism by applying it to some models for relativistic
particles.
