On field theoretic generalizations of a Poisson algebra

Geometry of Hamiltonian n-vector fields in multisymplectic field theory

“…to formulate the equations of motion but it is useful to consider functions like the DW Hamiltonian H without evaluating them on fields ϕ(x), etc. To this end, let us introduce coordinates v A for fields, v A µ for their spacetime derivatives and p µ A for the polymomenta functions (7). To condense notation, we will write derivatives w.r.t.…”

“…While Forger and Römer [4] work on the extended multisymplectic phase space P that generalises the doubly extended phase space of time dependent symplectic mechanics, Kanatchikov [7,8] uses a space that has one dimension less than P and can be interpreted as the parameter space of hypersurfaces of constant DW Hamiltonians. This space will be denotedP for the rest of this paper.…”

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“…(1.1), have been studied in our previous papers [27][28][29] to which we refer for more details. The analogue of the Poisson bracket in the DW formulation is deduced from the object, called the polysymplectic form, which in local coordinates can be written in the form 1 Ω = −dy a ∧ dp µ a ∧ ω µ and is viewed as a field theoretic generalization of the symplectic form within the DW formulation.…”

“…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.…”

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“…VI a (2n + 1)-bracket is introduced and some of its general properties are proved. The bracket differs from the so-called generalized Nambu, generalized Poisson, or generalized Nambu-Poisson brackets discussed in the literature (Bayen and Flato, 1975;Cohen, 1975 Kanatchikov, 1997). The particular case of a 3-bracket dynamics is discussed in Sec.…”