  We present the Lagrangian whose corresponding action is the trace K action
for General Relativity. Although this Lagrangian is second order in the
derivatives, it has no second order time derivatives and its behaviour at space
infinity in the asymptotically flat case is identical to other alternative
Lagrangians for General Relativity, like the gamma-gamma Lagrangian used by
Einstein. We develop some elements of the variational principle for field
theories with boundaries, and apply them to second order Lagrangians, where we
stablish the conditions -- proposition 1 -- for the conservation of the Noether
charges. From this general approach a pre-symplectic form is naturally obtained
that features two terms, one from the bulk and another from the boundary. When
applied to the trace K Lagrangian, we recover a pre-symplectic form first
introduced using a different approach. We prove that all diffeomorphisms
satisfying certain restrictions at the boundary -- that keep room for a
realization of the Poincar\'e group -- will yield Noether conserved charges. In
particular, the computation of the total energy gives, in the asymptotically
flat case, the ADM result.
