On the Difference between Admissible and "Differentiable" Hamiltonians

TL;DR

This paper revises the boundary condition criteria for Hamiltonians in systems with noncanonical Poisson structures, using examples from gravity and fluid dynamics to illustrate the modifications needed.

Contribution

It introduces a modified boundary criterion for Hamiltonians with noncanonical Poisson structures, expanding the applicability of the Regge-Teitelboim approach.

Findings

Boundary contributions to Hamiltonian equations of motion must cancel.

Boundary variations of the Hamiltonian are permitted.

Applications to Ashtekar gravity and ideal fluid hydrodynamics.

Abstract

It is shown that the Regge-Teitelboim criterion for fixing the unique boundary contribution to the Hamiltonian compatible with free boundary conditions should be modified if the Poisson structure is noncanonical. The new criterion requires cancellation of boundary contributions to the Hamiltonian equations of motion. In the same time, boundary contributions to the variation of Hamiltonian are allowed. The Ashtekar formalism for gravity and hydrodynamics of the ideal fluid with a free surface in the Clebsch variables are treated as examples.