Olga Rossi: Lagrangian and Hamiltonian duality

Lagrangian and Hamiltonian formalism certainly belong to most useful and extensively used mathematical frameworks in physics. The relationship between these two theories is provided by the Legendre transformation, and this relationship is one-to-one when the Lagrangian is regular: in such a case the Legendre transformation is a local diffeomorphism. In the standard treatment of many important physical systems, like for example the degenerate Lagrangians of classical mechanics, Maxwell equations, Dirac equations, Yang-Mills equations, Einstein equations, and others, the symmetry between the Lagrangian and Hamiltonian side is broken, and the theory of integrable Hamiltonian systems does not have a corresponding Lagrangian counterpart. During the past sixty years several approaches have been suggested to overcome these difficulties, the best known and most frequently used being Dirac's Hamiltonian theory of constrained systems. Since then a lot of work has been done to apply, explain, understand, improve and generalize Dirac's approach. Nevertheless, even now this subject still raises questions to be answered, limiting progress in both classical and quantum dynamics of constrained Hamiltonian systems. In my talk I will present a different approach, based on the study of Lepage manifolds which extend symplectic and multisymplectic structures. This setting provides different coordinate independent covariant Hamilton equations associated to given Euler-Lagrange equations, and, behind the standard ones, new regularity conditions and Legendre transformations. For many traditionally singular Lagrangians this approach provides a Hamiltonian system which is in a direct correspondence with the Lagrangian system, avoiding the use of Dirac constraints, and providing, among others, a new “proper” Hamiltonian for the regularized field.