Hamiltonian formulation for the motion of vortices in the presence of a free surface for ideal flow

In this paper, the work of Zakharov [J. Appl. Mech. Tech. Phys., Engl. Transl. 2, 190 (1968)] giving the canonical-Hamiltonian formulation for irrotational ideal flow with a free surface is generalized to the case in which the fluid interior contains isolated vortices. In particular, for two-dimensional flow the case of point vortices in the fluid interior is considered and for three-dimensional flow the case of vortex filaments in the interior is considered. Canonical variables are obtained explicitly for each of these cases. In the idealization of infinitely thin filaments, one has the usual problem of an infinite normal velocity being induced on a curved filament, so that a finite thickness must be given to each filament. This procedure is handled naturally by the Hamiltonian formulation given.