Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1501.04604

Classification of Arnold-Beltrami Flows and their Hidden Symmetries

Pietro Fre·Alexander S. Sorin
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math-phhep-thmath.DSmath.GRmath.MPnlin.CD

Keep this discovery

BibTeXRIS

Pietro Fre, Alexander S. Sorin. 2015-01-19. Classification of Arnold-Beltrami Flows and their Hidden Symmetries. https://arxiv.org/abs/1501.04604

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Why we should condition denoising diffusion generative models on windows of past observations

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion ``direct observation prediction'' (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

math-ph↗
arXiv2026

The kinematic structures and the inertial geometry of a moving charge

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one --- a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

math-ph↗