Canonical Description of Ideal Magnetohydrodynamics and Integrals of Motion

In the framework of the variational principle there are introduced canonical variables describing magnetohydrodynamic (MHD) flows of general type without any restrictions for invariants of the motion. It is shown that the velocity representation of the Clebsch type introduced by means of the variational principle with constraints is equivalent to the representation following from the generalization of the Weber transformation for the case of arbitrary MHD flows. The integrals of motion and local invariants for MHD are under examination. It is proved that there exists generalization of the Ertel invariant. It is expressed in terms of generalized vorticity field (discussed earlier by Vladimirov and for the incompressible case). The generalized vorticity presents the frozen-in field for the barotropic and isentropic flows and therefore for these flows there exists generalized helicity invariant. This result generalizes one obtained by Vladimirov and Moffatt in the cited work for the incompressible fluid. It is shown that to each invariant of the conventional hydrodynamics corresponds MHD invariant and therefore our approach allows correct limit transition to the conventional hydrodynamic case. The additional advantage of the approach proposed enables one to deal with discontinuous flows, including all types of possible breaks.