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 so named ABC flows introduced by Beltrami, Arnold and Childress. Main reference points are Arnold’s theorem stating that, for flows taking place on compact three manifolds ℳ₃, the only velocity fields able to produce chaotic streamlines are those satisfying Beltrami equation and the modern topological conception of contact structures, each of which admits a representative contact one-form also satisfying Beltrami equation. We advocate that Beltrami equation is nothing else but the eigenstate equation for the first order Laplace-Beltrami operator ★ _g d, which can be solved by using time-honored harmonic analysis. Taking for ℳ₃, a torus T ³ constructed as ℝ³/Λ, where Λ is a crystallographic lattice, we present a general algorithm to construct solutions of the Beltrami equation which utilizes as main ingredient the orbits under the action of the point group B _A of three-vectors in the momentum lattice *Λ. Inspired by the crystallographic construction of space groups, we introduce the new notion of a Universal Classifying Group $\mathfrak{G}\mathfrak{A}_\Lambda$ which contains all space groups as proper subgroups. We show that the ★ _g d eigenfunctions are naturally arranged into irreducible representations of $\mathfrak{G}\mathfrak{A}_\Lambda$ and by means of a systematic use of the branching rules with respect to various possible subgroups $H_i \subset \mathfrak{G}\mathfrak{A}_\Lambda$ we search and find Beltrami fields with non trivial hidden symmetries. In the case of the cubic lattice the point group is the proper octahedral group O₂₄ and the Universal Classifying Group $\mathfrak{G}\mathfrak{A}_{cubic}$ is a finite group G₁₅₃₆ of order |G₁₅₃₆| = 1536 which we study in full detail deriving all of its 37 irreducible representations and the associated character table. We show that the O₂₄ 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 G₁₅₃₆. 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 need of a field-theory yielding Beltrami equation as a field equation and/or an instanton equation and on 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 are different from the non-linear one proposed by Arnold and possibly make contact with M-theory and the geometry of flux-compactifications.