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Multisymplectic structures and the variational bicomplex

Published online by Cambridge University Press:  04 August 2009

THOMAS J. BRIDGES ,
PETER E. HYDON  and
JEFFREY K. LAWSON
THOMAS J. BRIDGES
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU 2 7XH. e-mail: T.Bridges@surrey.ac.uk, P.Hydon@surrey.ac.uk
PETER E. HYDON
Affiliation:
Department of Mathematics, University of Surrey, Guildford, Surrey GU 2 7XH. e-mail: T.Bridges@surrey.ac.uk, P.Hydon@surrey.ac.uk
JEFFREY K. LAWSON
Affiliation:
Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC 28723, U.S.A.
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Abstract

Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 148 , Issue 1 , January 2010 , pp. 159 - 178
Copyright
Copyright © Cambridge Philosophical Society 2009

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