Periodic solutions of singular first-order Hamiltonian systems of N-vortex type

We are concerned with the dynamics of N point vortices z(1), ..., z(N) is an element of Omega subset of R-2 in a planar domain. This is described by a Hamiltonian system Gamma(k)(z)over dot(k)(t) = J del(zk) H(z(t)), k = 1, ..., N, where Gamma(1), ..., Gamma(N) is an element of R \ {0} are the vorticities, J is an element of R-2x2 is the standard symplectic 2 x 2 matrix, and the Hamiltonian H is of N-vortex type: H(z1, ..., zN) = -1/2 pi Sigma(N)(j,k=1j not equal k) Gamma(j)Gamma(k) log vertical bar z(j) - z(k)vertical bar - Sigma(N)(j,k=1) Gamma(j)Gamma(k)g(z(j), z(k)). Here g : Omega x Omega -> R is an arbitrary symmetric function of class C-2, e.g., the regular part of a hydrodynamic Green function. Given a non-degenerate critical point a(0) is an element of Omega of h(z) = g(z, z) and a non-degenerate relative equilibrium Z(t) is an element of R-2N of the Hamiltonian system in the plane with g = 0, we prove the existence of a smooth path of periodic solutions z((r))(t) = (z(1)((r))(t), ..., z(N)((r))(t)) is an element of Omega(N), 0 < r < r(0), with z(k)((r))(t) -> a(0) as r -> 0. In the limit r -> 0, and after a suitable rescaling, the solutions look like Z(t).