  The Lagrangian proposed by York et al. and the covariant first order
Lagrangian for General Relativity are introduced to deal with the (vacuum)
gravitational field on a reference background. The two Lagrangians are compared
and we show that the first one can be obtained from the latter under suitable
hypotheses. The induced variational principles are also compared and discussed.
A conditioned correspondence among Noether conserved quantities, quasilocal
energy and the standard Hamiltonian obtained by 3+1 decomposition is also
established. As a result, it turns out that the covariant first order
Lagrangian is better suited whenever a reference background field has to be
taken into account, as it is commonly accepted when dealing with conserved
quantities in non-asymptotically flat spacetimes. As a further advantage of the
use of a covariant first order Lagrangian, we show that all the quantities
computed are manifestly covariant, as it is appropriate in General Relativity.
