Second-order Lagrangians admitting a first-order Hamiltonian formalism

Second-order Lagrangian densities admitting a first-order Hamiltonian formalism are studied; namely, i) for each second-order Lagrangian den- sity on an arbitrary fibred manifold p: E ? N the Poincar¿e-Cartan form of which is projectable onto J1E, by using a new notion of regularity pre- viously introduced, a first-order Hamiltonian formalism is developed for such a class of variational problems; ii) the existence of first-order equiva- lent 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