  We continue a previous analysis of the covariant Hamiltonian symplectic
structure of General Relativity for spatially bounded regions of spacetime. To
allow for near complete generality, the Hamiltonian is formulated using any
fixed hypersurface, with a boundary given by a closed spacelike 2-surface. A
main result is that we obtain Hamiltonians associated to Dirichlet and Neumann
boundary conditions on the gravitational field coupled to matter sources, in
particular a Klein-Gordon field, an electromagnetic field, and a set of
Yang-Mills-Higgs fields. The Hamiltonians are given by a covariant form of the
Arnowitt-Deser-Misner Hamiltonian modified by a surface integral term that
depends on the particular boundary conditions. The general form of this surface
integral involves an underlying ``energy-momentum'' vector in the spacetime
tangent space at the spatial boundary 2-surface. We give examples of the
resulting Dirichlet and Neumann vectors for topologically spherical 2-surfaces
in Minkowski spacetime, spherically symmetric spacetimes, and stationary
axisymmetric spacetimes. Moreover, we show the relation between these vectors
and the ADM energy-momentum vector for a 2-surface taken in a limit to be
spatial infinity in asymptotically flat spacetimes. We also discuss the
geometrical properties of the Dirichlet and Neumann vectors and obtain several
striking results relating these vectors to the mean curvature and normal
curvature connection of the 2-surface. Most significantly, the part of the
Dirichlet vector normal to the 2-surface depends only the spacetime metric at
this surface and thereby defines a geometrical normal vector field on the
2-surface. Properties and examples of this normal vector are discussed.
