Singular manifolds and quasi-periodic solutions of Hamiltonians for vortex lattices

Abstract

A general method for establishing the existence of quasi-periodic solutions of Hamiltonian systems for vortex lattices is illustrated in a simple example involving two degrees of freedom. The geometry of intersecting singular manifolds of the Hamiltonians introduces suitable canonical transformations which put the Hamiltonian into the form of singular weakly coupled oscillators. As by-products of this procedure, additional integrals of motion are found for the leading term in the transformed Hamiltonian. These extra integrals are approximate invariants for the full Hamiltonians.