1998/09/18 by Vladimir O. Soloviev, Soloviev, Vladimir O.
Mathematics · Physics and Astronomy · #58F05 (Primary) 70G50 #58G20 (Secondary) #Algebraic structures and combinatorial models #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #hep-th #math-ph #math.DG #math.MP #msc:58F05 #msc:58G20 #msc:70G50
paper · pdf · doi:10.48550/arxiv.math/9809103
21 pages, Latex, amssymb.sty, twoside.sty. Talk given at the International Conference on Secondary calculus and cohomological Physics, Moscow, August 1997
arxiv created 1998/09/18 · openalex publication_date 1998/09/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is shown that the new formula for the field theory Poisson brackets arise naturally in the extension of the formal variational calculus incorporating divergences. The linear spaces of local functionals, evolutionary vector fields, functional forms, multi-vectors and differential operators become graded with respect to divergences. The bilinear operations, such as the action of vector fields on functionals, the commutator of vector fields, the interior product of forms and vectors and the Schouten-Nijenhuis bracket are compatible with the grading. A definition of the adjoint graded operator is proposed and antisymmetric operators are constructed with the help of boundary terms. The fulfilment of the Jacobi identity for the new Poisson brackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis bracket of the Poisson bivector with itself. It is demonstrated, as an example, that the second structure of the Korteweg-de Vries equation is not Hamiltonian with respect to the new brackets until special boundary conditions are prescribed.