Tulczyjew triples in the constrained dynamics of strings

TL;DR

This paper develops a covariant geometric framework using Tulczyjew triples for the constrained dynamics of strings, generalizing classical mechanics to higher exterior powers of tangent bundles and including nonholonomic constraints.

Contribution

It introduces a natural Tulczyjew triple for string dynamics using exterior powers of tangent bundles, extending the geometric approach to constrained string and static problems.

Findings

Derivation of phase and Euler-Lagrange equations in this framework

Inclusion of nonholonomic constraints in string dynamics

Application to constrained Plateau problem in statics

Abstract

We show that there exists a natural Tulczyjew triple in the dynamics of objects for which the standard kinematic configuration space $TM$, i.e. the tangent bundle, is replaced with its $n$-th exterior power, i.e. the bundle of tangent $n$-vectors. In this framework, which is fully covariant, we geometrically derive phase equations, as well as Euler-Lagrange equations, including nonholonomic constraints into the picture. Dynamics of strings and a constrained Plateau problem in statics are particular cases of this framework.