  The ADM formalism is reviewed and techniques for decomposing generic
components of metric, connection and curvature are obtained. These techniques
will turn out to be enough to decompose not only Einstein equations but also
covariant conservation laws. Then a number of independent sets of hypotheses
that are sufficient (though non-necessary) to obtain standard ADM quantities
(and Hamiltonian) from covariant conservation laws are considered. This
determines explicitely the range in which standard techniques are equivalent to
covariant conserved quantities. The Schwarzschild metric in different
coordinates is then considered, showing how the standard ADM quantities fail
dramatically in non-Cartesian coordinates or even worse when asymptotically
flatness is not manifest; while, in view of their covariance, covariant
conservation laws give the correct result in all cases.
