Boundary values as Hamiltonian variables. I. New Poisson brackets

The ordinary Poisson brackets in field theory do not fulfill the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. It is shown that these brackets can be modified by adding some surface terms to lift this restriction. The new brackets generalize a canonical bracket considered by Lewis, Marsden, Montgomery, and Ratiu for the free boundary problem in hydrodynamics. The definition of Poisson brackets used herein permits the treating of to treat boundary values of a field on equal footing with its internal values and the direct estimation of estimate the brackets between both surface and volume integrals. This construction is applied to any local form of Poisson brackets.