Clebsch Variational Principles in Field Theories and Singular Solutions of 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-Poincaré 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.

Matrix G-strands

“…Exploring that connection with the SE(3) G-Strand equations could be a promising direction for future research, particularly for the integrable cases. Of course, the G-Brane equations for the action of SE(3) on smooth embeddings Emb(M, R 3 ) for a smooth manifold M ⊂ R 3 would also be promising and the mathematical formulation for G-Branes has recently been discussed in [10]. Applications of similar ideas for registration of planar curves have recently been successful [22].…”

Invariant variational problems on homogeneous spaces

“…We recall the reduced equations for space-time strands g : R×M → G as in [4,5]. These are also called sliced covariant EP equations because the sliced manifold R × M is considered instead of M in the covariant EP equations.…”

“…The case when the manifold M is sliced leads to the reduced equations for space-time strands, also called sliced covariant EP equations [4,5]. Special cases are the G-strands [6].…”

“…We shall follow the approach of [3] based on affine EP reduction. See also [5] for more explanation of the link between G-strands, covariant EP, and affine EP reductions.…”

Multisymplectic variational integrators and space/time symplecticity

“…In the continuous case, the articles Marsden, and Shkoller [1999], Marsden, Pekarsky, Shkoller and West [2001], Fetecau, Marsden, West [2003], Yavari, Marsden, and Ortiz [2006], Ellis, Gay-Balmaz, Holm, Putkaradze, Ratiu [2010] are examples of papers in which multisymplectic geometry has been further developed and applied in the context of continuum mechanics. We refer to Castrillón-López, Ratiu, and Shkoller [2000], Castrillón-López and Ratiu [2003], Ellis, Gay-Balmaz, Holm, Ratiu [2011], Gay-Balmaz [2013] for the development and the use of the techniques of reduction by symmetries for covariant field theory.…”