Second-order Lagrangians admitting a first-order Hamiltonian formalism

Second-order Lagrangian densities admitting a first-order Hamiltonian formalism are studied; namely, (1) for each second-order Lagrangian density on an arbitrary fibred manifold p:E→N the Poincaré–Cartan form of which is projectable onto J1E, by using a new notion of regularity previously introduced, a first-order Hamiltonian formalism is developed for such a class of variational problems; (2) the existence of first-order equivalent Lagrangians is discussed from a local point of view as well as global; (3) this formalism is then applied to classical Hilbert–Einstein Lagrangian and a generalization of the BF theory. The results suggest that the class of problems studied is a natural variational setting for GR.