Multisymplectic geometry and covariant formalism for mechanical systems with a Lie group as configuration space: application to the Reissner beam

TL;DR

This paper develops a covariant multisymplectic geometric framework for mechanical systems with Lie group configuration spaces, deriving Euler-Poincaré equations and Noether currents, exemplified on the Reissner beam.

Contribution

It reconstructs the geometric reduction for covariant problems with Lie groups, providing explicit multisymplectic forms and equations for such systems.

Findings

Derived multisymplectic forms for principal G bundles

Formulated covariant Euler-Poincaré equations

Established Noether current forms in dual Lie algebra

Abstract

Many physically important mechanical systems may be described with a Lie group $G$ as configuration space. According to the well-known Noether's theorem, underlying symmetries of the Lie group may be used to considerably reduce the complexity of the problems. However, these reduction techniques, used without care for general problems (waves, field theory), may lead to uncomfortable infinite dimensional spaces. As an alternative, the \emph{covariant} formulation allows to consider a finite dimensional configuration space by increasing the number of independent variables. But the geometric elements needed for reduction, adapted to the specificity of covariant problems which admit Lie groups as configuration space, are difficult to apprehend in the literature (some are even missing to our knowledge). To fill this gap, this article reconsiders the historical geometric construction made by E. Cartan in this particular "covariant Lie group" context. Thus, and it is the main interest of this work, the Poincar\'e-Cartan and multi-symplectic forms are obtained for a principal $G$ bundle. It allows to formulate the \emph{Euler-Poincar\'e equations} of motion and leads to a Noether's current form defined in the dual Lie algebra.