Boundary values as Hamiltonian variables. II. Graded Structures

TL;DR

This paper extends variational calculus with divergences to naturally produce graded Poisson brackets, linking them to Hamiltonian structures and demonstrating their application to integrable systems like KdV.

Contribution

It introduces a graded extension of variational calculus that yields new Poisson brackets compatible with divergence-based structures.

Findings

New Poisson brackets arise naturally in graded variational calculus.

The Jacobi identity is linked to the Schouten-Nijenhuis bracket vanishing.

The second KdV structure is not Hamiltonian under the new brackets.

Abstract

It is shown that the new Poisson brackets proposed in Part I of this work (J. Math. Phys. 34, 5747(hep-th/9305133)) arise naturally in an 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. Bilinear operations, such as action of vector fields onto functionals, commutator of vector fields, 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 skew-adjoint operators are constructed with the help of boundary terms. Fulfilment of the Jacobi identity for the new Poisson brackets is shown to be equivalent to vanishing of the Schouten-Nijenhuis bracket for Poisson bivector with itself. The simple procedure for testing this condition proposed by P. Olver is applicable with a minimal modification. It is demonstrated that the second structure of the Korteweg--de Vries equation is not Hamiltonian with respect to the new brackets.