Clebsch Variational Principles in Field Theories and Singular Solutions of Covariant Epdiff Equations

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–Poincarë 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 new class of covariant Pontryagin variational principles in field theories.