Olga Rossi: Hamiltonian systems and connections

In this talk I consider systems of second-order partial differential equations arising as the variational equations of a regular Lagrangian.

I explain how the problem of generalizing the symplectic Hamilton equations on the cotangent bundle to the case of PDE's and field theory
can be solved. I will present very recent results obtained in collaboration with David Saunders. Our approach involves the use of connections related to the Hamiltonian systems. Main results concern the construction of connections satisfying the condition that every integral section is an extremal of the variational problem, and conversely that every extremal can be embedded in a connection of this kind.

We give an explicit description of all such connections, and in this way we can describe all the local solutions of the Euler--Lagrange equations in terms of the related connections. Application of these results to second order PDE's seems to provide a new method of generating solutions from a fixed (known) solution, and of constructing families of solutions of the PDE's with prescribed initial conditions.