  We investigate the covariant Hamiltonian symplectic structure of General
Relativity for spatially bounded regions of spacetime with a fixed time-flow
vector. For existence of a well-defined Hamiltonian variational principle
taking into account a spatial boundary, it is necessary to modify the standard
Arnowitt-Deser-Misner Hamiltonian by adding a boundary term whose form depends
on the spatial boundary conditions for the gravitational field. The most
general mathematically allowed boundary conditions and corresponding boundary
terms are shown to be determined by solving a certain equation obtained from
the symplectic current pulled back to the hypersurface boundary of the
spacetime region. A main result is that we obtain a covariant derivation of
Dirichlet, Neumann, and mixed type boundary conditions on the gravitational
field at a fixed boundary hypersurface, together with the associated
Hamiltonian boundary terms. As well, we establish uniqueness of these boundary
conditions under certain assumptions motivated by the form of the symplectic
current. Our analysis uses a Noether charge method which extends and unifies
several results developed in recent literature for General Relativity. As an
illustration of the method, we apply it to the Maxwell field equations to
derive allowed boundary conditions and boundary terms for existence of a
well-defined Hamiltonian variational principle for an electromagnetic field in
a fixed spatially bounded region of Minkowski spacetime.
