An integrable family of Monge-Ampere equations and their multi-Hamiltonian structure

We have identified a completely integrable family of Monge-Ampère equations through an examination of their Hamiltonian structure. Starting with a variational formulation of the Monge-Ampère equations we have constructed the first Hamiltonian operivtor through an application of Dirac’s theory of constraints. The completely integrable class of Monge-Ampère equations are then obtained by solving the .Jacobi identities for a sufficiently general form of the second Hamiltonian operator that is compatible with the first. Furthermore, Chern, Levine and Nirenberg have long ago pointed out the distinguished role that the complex homogeneous Monge-Ampère equation plays in the theory of functions of several complex variables. In particular Semmes has called attention to the symplectic structure of the geodesic flow defined by this equation. A new approach to this problem in the framework of dynamical .systems ( with infinitely many degrees of freedom ) shows that it is a completely integrable system. This example exhibits several new features in the theory of integrable systems as well. Namely it is an integrable system in arbitrary dimension and furthermore admits infinitely many symplectic structures. The latter is the key to a proof of integrability through Magri’s theorem which requires only bi-Hamiltonian structure.