Ideal hydrodynamics on Lie groups

In this paper we consider the reduction to a Lie group of geodesic flows of left- or right-invariant metrics: for a fixed value of momentum we describe the reduced vector field on the Lie group, as well as its (right or left-invariant) symmetry fields. The reduced vector field has an important hydrodynamic interpretation: it is a stationary flow of an ideal fluid with a constant pressure. We use the explicit expressions for the reduced vector field and its symmetry fields to define “secondary hydrodynamics”, i.e., we study the reduction for the Euler equations of an ideal fluid, that describe the geodesics of a right-invariant metric on a Lie group SDiff(M) of the volume-preserving diffeomorphisms of a Riemannian manifold MM, to the group SDiff(M). For a “typical” coadjoint orbit we find all symmetry fields of a reduced flow, and, as a corollary, we get a simple proof for nonexistence of new invariants of coadjoint orbits, which are the integrals of local densities over the flow domain.