A vertical exterior derivative in multisymplectic geometry and a graded poisson bracket for nontrivial geometries

Abstract

A vertical exterior derivative is constructed that is needed for a graded Poisson structure on multisymplectic manifolds over nontrivial vector bundles. In addition, the properties of the Poisson bracket are proved and first examples are discussed. *

Geometry of Hamiltonian n-vector fields in multisymplectic field theory

“…Eq. (14). The corresponding Hamiltonian forms are functions on the extended multisymplectic phase space P. If such a Hamiltonian function is of the special form…”

“…Moreover, the sense in which multivector fields are related to distributions seems to be folklore and is written out explicitly in the work by Echeverría-Enríquez et al [3], see Appendix A of this paper. However, both use the smaller multisymplectic phase spaceP which requires the choice of a connection [14]. Moreover, we will show in Theorem 3 that for typical cases in field theory the generalisation of (1) toP does not admit the interpretation of X f to define a distribution.…”

“…where VE is the vertical tangent subbundle of E and R denotes a trivial line bundle on E. For the latter isomorphism, a connection Γ of E is needed in addition [14]. Note that the tensor products are understood pointwise on E. At this point it is useful to examine the special case if M happens to be the real axis R, i.e.…”

Untitled

“…( * is the Hodge duality operator), with respect to which the space of Hamiltonian forms is stable [28][29][30].…”

“…It is only recently that a proper Poisson bracket operation, which is defined on differential forms representing the dynamical variables and leads to a Poisson-Gerstenhaber algerba structure, has been found within the DW theory in [25][26][27][28][29] (see also [18,30,31] for recent generalizations). This progress has been accompanied and followed by further developments in "multisymplectic" generalizations of the symplectic geometry aimed at applications in field theory and the calculus of variations [38,39,[41][42][43][44] and in other geometric aspects of the Lagrangian and Hamiltonian formalism in field theory [32-37, 40, 45] which to a great extent are been so far basically ignored by the wider mathematical physics community.…”