Frenet–Serret dynamics

Abstract

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 Poincaré 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 Poincaré 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.