Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Multisymplectic structures and the variational bicomplex

THOMAS J. BRIDGESa1, PETER E. HYDONa1 and JEFFREY K. LAWSONa2

a1 Department of Mathematics, University of Surrey, Guildford, Surrey GU 2 7XH. e-mail: T.Bridges@surrey.ac.uk, P.Hydon@surrey.ac.uk

a2 Department of Mathematics and Computer Science, Western Carolina University, Cullowhee, NC 28723, U.S.A.

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.

(Received February 20 2009)

(Revised June 15 2009)

(Online publication August 04 2009)