Classification of Arnold-Beltrami Flows and their Hidden Symmetries

Abstract

In the context of mathematical hydrodynamics, we consider the group theory structure which underlies the ABC-flow introduced by Beltrami, Arnold and Childress. Beltrami equation is the eigenstate equation for the first order Laplace-Beltrami operator *d, which we solve by using harmonic analysis. Taking torus T^3 constructed as R^3/L, where L is a crystallographic lattice, we present a general algorithm to construct solutions of Beltrami equation which utilizes as main ingredient the orbits under the action of the point group P_L of three-vectors in the momentum lattice L*. We introduce the new notion of a Universal Classifying Group GU_L which contains all crystallographic space groups as proper subgroups. We show that the *d-eigenfunctions are naturally arranged into irreducible representations of GU_L and by means of a systematic use of the branching rules with respect to various possible subgroups H of GU_L we search and find Beltrami fields with non trivial hidden symmetries. In the case of the cubic lattice the point group P_L is the proper octahedral group O_24 and the Universal Classifying Group is finite group G_1536 of order 1536 which we study in full detail deriving all of its 37 irreducible representations and the associated character table. We show that the O_24 orbits in the cubic lattice are arranged into 48 equivalence classes, the parameters of the corresponding Beltrami vector fields filling all the 37 irreducible representations of G_1536. In this way we obtain an exhaustive classification of all generalized ABC-flows and of their hidden symmetries. We make several conceptual comments about the possible relation of Arnold-Beltrami flows with (supersymmetric) Chern-Simons gauge theories. We also suggest linear generalizations of Beltrami equation to higher odd-dimensions that possibly make contact with M-theory and the geometry of flux-compactifications.