  Second-order Lagrangian densities admitting a first-order Hamiltonian
formalism are studied; namely, i) for each second-order Lagrangian density on
an arbitrary fibred manifold $p\colon E\to N$ the Poincar\'e-Cartan form of
which is projectable onto $J^1E$, by using a new notion of regularity
previously introduced, a first-order Hamiltonian formalism is developed for
such a class of variational problems; ii) the existence of first-order
equivalent Lagrangians are discussed from a local point of view as well as
global; iii) this formalism is then applied to classical Einstein-Hilbert
Lagrangian and a generalization of the BF theory. The results suggest that the
class of problems studied is a natural variational setting for GR.
