Clebsch variational principles in field theories and singular solutions of covariant EPDiff equations

TL;DR

This paper develops a covariant field theoretic version of the Clebsch variational principle, providing new insights into covariant Euler-Poincaré equations and singular solutions like peakons and filaments in covariant EPDiff equations.

Contribution

It introduces a covariant Clebsch variational principle for field theories, linking it to covariant Euler-Poincaré equations and singular solutions, expanding the theoretical framework of multisymplectic field theories.

Findings

Derived covariant Euler-Poincaré equations using the new principle.

Provided a new interpretation of singular peakon solutions.

Constructed singular solutions such as filaments and sheets for covariant EPDiff equations.

Abstract

This paper introduces and studies a field theoretic analogue of the Clebsch variational principle of classical mechanics. This principle yields an alternative derivation of the covariant Euler-Poincar\'e equations that naturally includes covariant Clebsch variables via multisymplectic momentum maps. In the case of diffeomorphism groups, this approach gives a new interpretation of recently derived singular peakon solutions of Diff(R)-strand equations, and allows for the construction of singular solutions (such as filaments or sheets) for a more general class of equations, called covariant EPDiff equations. The relation between the covariant Clebsch principle and other variational principles arising in mechanics and field theories, such as Hamilton-Pontryagin principles, is explained through the introduction of a class of covariant Pontryagin variational principles in field theories.