Geodesics on extensions of Lie groups and stability; the superconductivity equation

The equations of motion of a charged ideal fluid, respectively the superconductivity equation (both in a given magnetic field) are showed to be geodesic equations on a general, respectively central extension of the group of volume preserving diffeomorphisms with right invariant metric. For this, quantization of the magnetic flux is required. We do curvature computations in both cases in order to get some informations about the stability.