Infinite Dimensional Hamiltonian Systems

where H is the Hamiltonian (”energy”) and {. , .} is a Poisson bracket on an infinite dimensional phase space, called Poisson manifold. Unlike finite dimensional Hamiltonian systems, which are ordinary differential evolution equations on finite dimensional phase spaces, for which general existence and uniqueness theorems for solutions exist, this is not the case for PDEs. There are no general existence and uniqueness theorems for solutions of infinite dimensional Hamiltonian systems. These have to be established case by case. In this Encyclopedia we can only give the broad mathematical frame work of infinite dimensional Hamiltonian systems. We shall give precise definitions and illustrate this concept on physical examples.