On variational features of vortex ows

Ideal incompressible ‡uid is a Hamiltonian system which possesses an in…nite number of inte-grals, the circulations of velocity over closed ‡uid contours. This allows one to split all the degrees of freedom into the driving ones and the "slave" ones, the latter to be determined by the integrals of motions. The "slave" degrees of freedom correspond to "potential part" of motion, which is driven by vorticity. Elimination of the "slave" degrees of freedom from equations of ideal incom-pressible ‡uid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle for this system was found recently (Berdichevsky, 1997, Kuznetsov and Ruban, 1998). It looks striking, however. In particular, the ‡uid motion is set to be compressible, while in the least action principle of ‡uid mechanics the incompressibility of motion is a built-in property. This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational principles are proposed for steady vortex ‡ows. Their relation to Arnold's variational principle of steady vortex motion is discussed.