Variational principle in canonical variables, Weber transformation and complete set of the local integrals of motion for dissipation–free magnetohydrodynamics

The intriguing problem of the «missing» MHD integrals of motion is solved in the paper, i.e., analogs of the Ertel, helicity and vorticity invariants are obtained. The two latter were discussed earlier in the literature only for the specific cases, and Ertel invariant is first presented. The set of ideal MHD invariants obtained appears to be complete: to each hydrodynamic invariant corresponds its MHD generalization. These additional invariants are found by means of the fluid velocity decomposition based on its representation in terms of generalized potentials. This representation follows from the discussed variational principle in Hamiltonian (canonical) variables and it naturally decomposes the velocity field into the sum of «hydrodynamic» and «magnetic» parts. The «missing» local invariants are expressed in terms of the «hydrodynamic» part of the velocity and therefore depend on the (non–unique) velocity decomposition, i.e., they are gauge–dependent. Nevertheless, the corresponding conserved integral quantities can be made decomposition–independent by the appropriate choosing of the initial conditions for the generalized potentials. It is also shown that the Weber transformation of MHD equations (partially integration of the MHD equations) leads to the velocity representation coinciding with that following from the variational principle with constraints. The necessity of exploiting the complete form of the velocity representation in order to deal with general–type MHD flows (non–barotropic, rotational and with all possible types of breaks as well) in terms of single–valued potentials is also under discussion. The new basic invariants found allows one to widen the set of the local invariants on the basis of the well–known recursion procedure.