Frenet-Serret dynamics

TL;DR

This paper develops a geometric framework for particle dynamics using Frenet-Serret curvatures, revealing integrability properties of models based on these curvatures in Minkowski space.

Contribution

It introduces a theory of deformations for FS frames and expresses Poincaré invariants in terms of FS curvatures, demonstrating integrability of certain higher-order models.

Findings

The first FS curvature model is integrable.

Poincaré invariants are simply expressed via FS curvatures.

Higher-order models, including second FS curvature, are also integrable in 3D.

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 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.