Boundary values as Hamiltonian variables. I. New Poisson brackets

TL;DR

This paper introduces modified Poisson brackets that include boundary surface terms, allowing boundary values to be treated on equal footing with internal values in field theory, thus addressing Jacobi identity issues.

Contribution

It proposes a new class of Poisson brackets with surface terms, generalizing previous brackets and enabling boundary values to be incorporated naturally in Hamiltonian formulations.

Findings

Modified brackets satisfy Jacobi identity with boundary conditions

Allows direct estimation of brackets between surface and volume integrals

Applicable to any local form of Poisson brackets

Abstract

The ordinary Poisson brackets in field theory do not fulfil the Jacobi identity if boundary values are not reasonably fixed by special boundary conditions. We show 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. Our definition of Poisson brackets permits to treat boundary values of a field on equal footing with its internal values and directly estimate the brackets between both surface and volume integrals. This construction is applied to any local form of Poisson brackets. A prescription for delta-function on closed domains and a definition of the {\it full} variational derivative are proposed.