Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems

Vortex line and magnetic line representations are introduced for a descript ion of flows in ideal hydrodynamics and magnetohydrodynamics (MHD), respect ively. For incompressible fluids, it is shown with the help of this transfo rmation that the equations of motion for vorticity Omega and magnetic field follow from a variational principle. By means of this representation, it i s possible to integrate the hydrodynamic type system with the Hamiltonian H =integral\Omega\dr and some other systems. It is also demonstrated that the se representations allow one to remove from the noncanonical Poisson bracke ts, defined in the space of divergence-free vector fields, the degeneracy c onnected with the vorticity frozenness for the Euler equation and with magn etic field frozenness for ideal MHD. For MHD, a new Weber-type transformati on is found. It is shown how this transformation can be obtained from the t wo-fluid model when electrons and ions can he considered as two independent fluids. The Weber-type transformation for ideal MHD gives the whole Lagran gian vector invariant. When this invariant is absent, this transformation c oincides with the Clebsch representation analog introduced by V.E. Zakharov and E. A. Kuznetsov [Dokl. Ajad. Nauk 194, 1288 (1970) [Sov. Phys. Dokl. 1 5, 913 (1971)]]. PACS number(s): 52.30.-9, 52.35.Ra, 52.55.Fa.