  We consider the motion of a particle described by an action that is a
functional of the Frenet-Serret [FS] curvatures associated with the embedding
of its worldline in Minkowski space. We develop a theory of deformations
tailored to the FS frame. Both the Euler-Lagrange equations and the physical
invariants of the motion associated with the Poincar\'e symmetry of Minkowski
space, the mass and the spin of the particle, are expressed in a simple way in
terms of these curvatures. The simplest non-trivial model of this form, with
the lagrangian depending on the first FS (or geodesic) curvature, is
integrable. We show how this integrability can be deduced from the Poincar\'e
invariants of the motion. We go on to explore the structure of these invariants
in higher-order models. In particular, the integrability of the model described
by a lagrangian that is a function of the second FS curvature (or torsion) is
established in a three dimensional ambient spacetime.
