1993/12/20 by Igor V. Kanatchikov, Kanatchikov, Igor V.
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #hep-th
paper · pdf · doi:10.48550/arxiv.hep-th/9312162
LATEX, 30 pages, PITHA 93/41
arxiv created 1993/12/20 · openalex publication_date 1993/12/20 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The analogue of the Poisson bracket for the De Donder-Weyl (DW) Hamiltonian formulation of field theory is proposed. We start from the Hamilton- Poincaré-Cartan (HPC) form of the multidimensional variational calculus and define the bracket on the differential forms over the space-time (=horizontal forms). This bracket is related to the Schouten-Nijenhuis bracket of the multivector fields which are associated with the horizontal forms by means of the "polysymplectic form". The latter is given by the HPC form and generalizes the symplectic form to field theory. We point out that the algebra of forms with respect to our Poisson bracket and the exterior product has the structure of the Gerstenhaber graded algebra. It is shown that the Poisson bracket with the DW Hamiltonian function generates the exterior differential thus leading to the bracket representation of the DW Hamiltonian field equations. Few illustrative examples are also presented.