Canonical multisymplectic structure on the total exterior algebra bundle
Abstract
The aim of this paper is to construct multisymplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an ndimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M. On the fibre, which has dimension 2^{n}, the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these nsymplectic structures define a partial differential operator, J_{∂}, which turns out to be a Dirac operator with multisymplectic structure. The operator J_{∂} generalizes the product operator J(d/dt) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J_{∂} is elliptic, and the generalization of Hamiltonian systems, J_{∂}Z=∇S(Z), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S(Z) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on kforms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with nontrivial curvature. Some applications and implications of the theory are also discussed.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 May 2006
 DOI:
 10.1098/rspa.2005.1629
 Bibcode:
 2006RSPSA.462.1531B
 Keywords:

 symplecticity;
 differential forms;
 nonlinear elliptic PDEs;
 Dirac operators