Hamiltonian structure for vortex filament flows

A new Hamiltonian formulation of the passive particle motion induced by a s mooth vortex filament in an ideal fluid contained in a region of 3-space is derived. The point of departure in the derivation is a desingularized vers ion of the Biot-Savart formula for the induced velocity field. Then a folia tion of a neighborhood of the filament (that moves with the fluid flow) is constructed that is comprised of smooth two-dimensional leaves that are inv ariant with respect to the induced velocity field at each time. Natural sym plectic coordinates are introduced on the moving leaves of the associated f oliation. such that the equations of motion on the leaves assume a simple ( possibly time-dependent) Hamiltonian form. With this Hamiltonian structure one can, by simply following the evolution of the leaves of the foliation, easily determine the motion of the passive fluid particles near the filamen t. Any irregular or singular behavior in the motion can essentially be asso ciated to geometrical features of the moving foliation in the large. The Ha miltonian structure is illustrated with three examples: a rectilinear filam ent; a circular vortex ring; and a helical filament.