Invariants of the Euler equations for ideal or barotropic hydrodynamics and superconductivity in D dimensions

Abstract

The Hamiltonian formalism for the Euler equations of an ideal fluid, superconductivity and a barotropic fluid on a D-dimensional Riemannian manifold is proposed. We show that each of these equations has an infinite series of integrals if D is even (“generalized enstrophies”) and at least one integral if D is odd (“generalized helicity”). We prove that the magnetic hydrodynamics integral f(v, B)μ is equal to the average linking number of vector fields rot v and B in terms of the ergodic theory. All the invariants considered are Casimir elements (i.e. invariants of coadjoint action) of the corresponding infinite-dimensional Lie algebras.