  We consider a general, classical theory of gravity with arbitrary matter
fields in $n$ dimensions, arising from a diffeomorphism invariant Lagrangian,
$\bL$. We first show that $\bL$ always can be written in a ``manifestly
covariant" form. We then show that the symplectic potential current
$(n-1)$-form, $\th$, and the symplectic current $(n-1)$-form, $\om$, for the
theory always can be globally defined in a covariant manner. Associated with
any infinitesimal diffeomorphism is a Noether current $(n-1)$-form, $\bJ$, and
corresponding Noether charge $(n-2)$-form, $\bQ$. We derive a general
``decomposition formula" for $\bQ$. Using this formula for the Noether charge,
we prove that the first law of black hole mechanics holds for arbitrary
perturbations of a stationary black hole. (For higher derivative theories,
previous arguments had established this law only for stationary perturbations.)
Finally, we propose a local, geometrical prescription for the entropy,
$S_{dyn}$, of a dynamical black hole. This prescription agrees with the Noether
charge formula for stationary black holes and their perturbations, and is
independent of all ambiguities associated with the choices of $\bL$, $\th$, and
$\bQ$. However, the issue of whether this dynamical entropy in general obeys a
``second law" of black hole mechanics remains open. In an appendix, we apply
some of our results to theories with a nondynamical metric and also briefly
develop the theory of stress-energy pseudotensors.
