Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type

The paper deals with singular first order Hamiltonian systems of the form where defines the standard symplectic structure in , are given, and the Hamiltonian H is of N-vortex type: This is defined on the configuration space of N different points in the domain . The function may have additional singularities near the boundary of . We prove the existence of a global continuum of periodic solutions that emanates, after introducing a suitable singular limit scaling, from a relative equilibrium of the N-vortex problem in the whole plane (where ). Examples for Z include Thomson's vortex configurations, or equilateral triangle solutions. The domain need not be simply connected. A special feature is that the associated action integral is not defined on an open subset of the space of -periodic functions, the natural form domain for first order Hamiltonian systems. This is a consequence of the singular character of the Hamiltonian. Our main tool in the proof is a degree for -equivariant gradient maps that we adapt to this class of potential operators.