Variational principles for spin systems and the Kirchhoff rod

TL;DR

This paper develops a geometric framework using variational principles and Euler-Poincaré equations to describe the dynamics of continuum spin systems and Kirchhoff's rod, unifying their mathematical treatment.

Contribution

It introduces a unified geometric approach to derive equations of motion for spin systems and Kirchhoff's rod via affine Euler-Poincaré equations and Clebsch constraints.

Findings

Derived equations of motion for continuum spin systems

Unified geometric interpretation for spin systems and Kirchhoff's rod

Established a variational principle framework for these systems

Abstract

We obtain the affine Euler-Poincar\'e equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.