Poisson brackets with divergence terms in field theories: two examples

Abstract

In field theories one often works with the functionals which are integrals of some densities. These densities are defined up to divergence terms (boundary terms). A Poisson bracket of two functionals is also a functional, i.e., an integral of a density. Suppose the divergence term in the density of the Poisson bracket be fixed so that it becomes a bilinear form of densities of two functionals. Then the left-hand side of the Jacobi identity written in terms of densities is not necessarily zero but a divergence of a trilinear form. The question is: what can be said about this trilinear form, what kind of a higher Jacobi identity (involving four fields) it enjoys? Two examples whose origin is the theory of integrable systems are given.