Vortex line representation for the hydrodynamic type equation

In this paper we give a brief review of the recent results obtained by the author and his co-authors for description of three-dimensional vortical incompressible flows in the hydrodynamic type systems. For such flows we introduce a new mixed Lagrangian-Eulerian description- the so called vortex line representation (VLR), which corre-sponds to transfer to the curvilinear system of coordinates moving together with vortex lines. Introducing the VLR allows to establish the role of the Cauchy invariants from the point of view of the Hamiltonian description. In particular, these (Lagrangian) invariants, characterizing the property of frozenness of the generalized vorticity into fluids, are shown to represent the infinite (continuous) number of Casimirs for the so-called non-canonical Poisson brackets. The VLR allows to integrate partially the equations of motion, to exclude the infinite degeneracy due to frozenness of the prim-itive Poisson brackets and to establish in new variables the variational principle. It is shown that the original Euler equations for vortical flows coincides with the equations of motion of a charged compressible fluid moving due to a self-consistent electro-magnetic field. Transition to the Lagrangian description in a new hydrodynamics is equivalent to the VLR. The VLR, as a mapping, turns out to be compressible that gives a new opportunity for collapse in fluid systems- breaking of vortex lines, result-ing in infinite vorticity. It is shown that such process is possible for three-dimensional integrable hydrodynamics with the Hamiltonian H = ∫ |Ω|dr where Ω is the vorticity. We also discuss some arguments in the favor of existence of such type of collapses for the Euler hydrodynamics, based on the results of some numerics.