Abstract
A gauge-free description of magnetohydrodynamic flows of an ideal incompressible fluid, which takes into account the freezing-in of the magnetic field and the presence of cross invariants containing the vorticity, is obtained. This description is an extension of the canonical formalism well-known in ordinary hydrodynamics to the dynamics of frozen-in flux lines. Magnetohydrodynamics is studied as the long-wavelength limit of the two-fluid model of a plasma, in which the existence of two frozen-in fields — curls of the generalized momenta of the electron and ion fluids — follows from the symmetry of each component with respect to relabeling of the Lagrangian labels. The cross invariants in magnetohydrodynamics are limits of special combinations of topological invariants of the two-fluid model. A variational principle is formulated for the dynamics of frozen-in magnetic flux lines, and the Casimir functionals of the noncanonical Poisson brackets are found.
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