Difference between admissible and “differentiable” Hamiltonians

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. At the same time, boundary contributions to the variation of the Hamiltonian are allowed. The Ashtekar formalism for gravity and hydrodynamics of the ideal fluid with a free surface in the Clebsch variables is treated as an example.