The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory

Abstract

We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.

The Notion of Observable in the Covariant Hamiltonian Formalism for the Calculus of Variations with Several Variables

“…If M = Λ n T * N and if H is a Legendre image Hamiltonian then any (n − 1)-form F of the type F = ξ θ, where ξ is a vector field on N , is algebraic observable (see [14], [8]). But as pointed out in Section 2.2 (definition 2.4) since L H m ⊂ KerdΠ m any vector field ζ which is a section of L H is necessarily "vertical" and satisfies ζ θ = 0.…”

Multisymplectic and Polysymplectic Structures on Fiber Bundles

“…One of them concerns the structure of the underlying Lie group: interpreting these structures as G-structures, what is the nature of G? Other questions refer to the definition of Poisson brackets (see the discussion in Refs [5][6][7]21]), the definition of actions of Lie groups and, more generally, of Lie groupoids on polysymplectic/polylagrangian or multisymplectic/multilagrangian fiber bundles, the construction of a corresponding momentum map (which would provide a general framework for the construction of Noether currents and the energy-momentum tensor within a direct and manifestly covariant hamiltonian approach), the formulation of a Marsden-Weinstein reduction procedure and, last but by no means least, the explicit construction of other classes of examples, in particular, analogues of the coadjoint orbit construction of symplectic geometry. All these problems are completely open and certainly will be the subject of much research in the future.…”

“…7 It follows that passage to the symbol can be regarded as a projection, from the space r s W * of (r − s)-horizontal r-forms on W to the space s V * ⊗ r−s T * of s-forms on V with values in the space 6 The first few terms of this sequence may be trivial, since…”

“…However, unlike in the symplectic case, these conditions alone are far too weak to guarantee the validity of a Darboux theorem, even at the purely algebraic level. For certain purposes, they may be sufficient to derive results that are of interest (for an example, see Refs [5][6][7]), but this version of the definition of a multisymplectic structureeven though often adopted in the literature, mostly for lack of a better one -is clearly inadequate. What is needed is an additional algebraic condition.…”

A general construction of poisson brackets on exact multusymplectic manifolds

“…A calculation in coordinates shows the last statement. The details of the proofs are contained in [3]. ✷ Remark.…”