Multisymplectic structures and the variational bicomplex

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.

On presymplectic structures for massless higher-spin fields

“…As is shown in [8,9], such a current defines a weak Lagrangian. This is given by a top form L on N built of the fields φ i and their derivatives.…”

“…Following [8,9], we say that a (2, n − 1)-form Ω on × N defines a presymplectic current compatible with equations of motion (3.15) if…”

“…So far such a correspondence has been established indirectly just by comparing tensor parameters entering the conserved currents and the symmetry transformations of the free HS fields. Furthermore, given a presymplectic structure, it is possible to define a local functional of fields whose stationary surface includes all the solutions to the original field equations [8,9]. It is hoped that function-als of this kind may find applications in the context of the AdS/CFT correspondence [10][11][12][13].…”

Emergence of unsteady dark solitary waves from coalescing spatially periodic patterns

“…The skew-symmetry of M assures that the PDE is conservative. Indeed, this system is an example of a multi-symplectic Hamiltonian PDE (Bridges 1997a,b;Bridges et al 2010). Here, the only property of the time-derivative term that will be important is the skew-symmetry of M. Examples that can be expressed in the form (1.6) are the nonlinear beam equation, the NLS equation, the good Boussinesq equation and the coupled-mode equation.…”

Multi-symplectic formulation of near-local Hamiltonian balanced models

“…(Note: the above construction may be extended to deal with systems for which W j , K a j and S also depend upon t and m, but this is not needed for our purposes-see Bridges et al (2010) for details. )…”