Dual Lagrangian field theories

We investigate how, under suitable regularity conditions, first-order Lagrangian field theories can be recasted in terms of a second-order Lagrangian, called the dual Lagrangian of the theory, depending on canonical conjugate momenta together with their derivatives. The necessary and sufficient conditions which allow such a (local) reformulation, obtained through a suitable generalization of the Legendre transformation, are analyzed. The global geometric framework is also investigated in detail. As an example, we apply the dual Lagrangian formulation to the Hilbert Lagrangian and to Euclidean self-dual gravity.