Symplectic and Poisson Geometry on Loop Spaces of Manifolds and Nonlinear Equations

We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian or multihamiltonian representations for some important nonlinear equations of mathematical physics and field theory such as nonlinear sigma models with torsion, degenerate Lagrangian systems of field theory, systems of hydrodynamic type, N-component systems of Heisenberg magnet type, Monge-Ampère equations, the Krichever-Novikov equation and others. In particular, complete classification of all nondegenerate Poisson bivectors ω(x, u, ux, uxx, ...) depending on derivatives of the field variables u(x) and the independent space variable x is obtained (u, i = 1, ..., N, are local coordinates on smooth manifold M). In other words, all Poisson brackets of the following form