An example is given in UFT256, curl q = kappa q, the curl is proportional to q, so is parallel to it, kappa being the scalar magnitude of wavevector. This was first used by Eugenio Beltrami in the eighteen eighties with curl v = alpha v, where v is velocity and alpha a scalar. It is Eq. (5) of

Donald Reed, pp. 525 ff. of M. W. Evans Ed., “Modern Nonlinear Optics”, volume 119(3) of “Advances in Chemical Physics”, (Wiley Interscience, New York, 2001), second edition.

This is available in all good libraries and there ought to be several libraries in the Munich area which have it. It can be purchased from Amazon as a hardback or e book. I am just about to write up Note 257(2) which will explain things in all detail. Beltrami flow has no Magnus force, and its helicoidal structure was discovered by Beltrami. It is illustrated in Figure 3 of Reed’s article, which also illustrates real Beltrami flows. Helicoidal means that the field lines start as transverse and end up as longitudinal. It is an axisymmetric sheared vortex. The B(3) field is related to the longitudinal field line. We have:

curl B(1) sub L = kappa B(1) sub L = curl B(2) sub L

curl B(1) sub R = – kappa B(1) sub R = curl B(2) sub R

curl B(3) = 0 B(3)

The curl eigenvalues are 1, 0, -1 . These are also the eigenvalues of helicity in quantum field theory. In UFT256 these were related directly to the spin connection of Cartan geometry. Here subscript L denotes anticlockwise rotating, subscript R denotes clockwise rotating. Beltrami flow is known loosely as “eigenfunctions of the curl operator”. Moses (cited by Reed) has shown that any vector field can be decomposed into (1), (2) and (3) modes (not just (1) and (2), a point of key importance because it infers the B(3) field immediately). In electrodynamics these lead to the B(1), B(2) and B(3) fields (H. F. Moses, SIAM J. Applied Mech., 21(1), 114 (1971)). This leads to the B Cyclic Theorem of O(3) electrodynamics:

B(1) x B(2) = i B(0) B(3)*

et cyclicum

which I inferred in the mid nineties independently of Moses and Reed. The formalism can also be related to Hamilton’s quaternions and to Cartan’s spinors in SU(2) rep., homomorphic with O(3) rep. So there is plenty of mileage here. The new ingredient post UFT256 is the spin connection. This is recognized now as being related to the alpha factor of Beltrami hydrodynamics, aerodynamics, electrodynamics, and possibly galactic dynamics. I have reviewed Beltrami electrodynamics in UFT100 Section 7, and in another UFT paper post UFT100. That can be found with google or keywords. Note that there is a small typo in UFT100 Section 7, the correct equations are as above.

Sent: 29/01/2014 07:42:05 GMT Standard Time

Subj: Re: Extract from Paper by Donald Reed in ACP vol 119(3), Modern Nonlinear Optics

Myron, Doug,

The review article is good, nethertheless I have an understanding problem with interpretation of the Beltrami equation. There are interesting diagrams how such a flow can be imagined, but to my understanding the curl of a vector is always perpendicular to the vector itself. This seems also to be the case in the longitudinal flux field examples. How can the curl be parallel to the vector?

Horst

EMyrone@aol.com hat am 28. Januar 2014 um 18:53 geschrieben:

Many thanks, this is an important review by Donald Reed.

To: EMyrone@aol.com

Sent: 28/01/2014 13:45:47 GMT Standard Time

Subj: Re: Fwd: Beltrami fields Forgot the link – Norman

I’ll send the link again . I think this is from the Reed paper you refered to . Here is the bit on B3 Regards Norman.

“Evans/Vigier Longitudinal B(3) Field and Trkalian Vector Fields

Developed over the past decade, concurrently with both Hillion/Quinnez and Rodrigues/Vaz SU(2) EM field models, but based upon a different non-Abelian gauge group, is the so-called Evans/Vigier longitudinal B(3) field representation [89-93]. In this model, a Yang-Mills gauge field theory [94] with an internal O(3) gauge field symmetry [95] is invoked to account for various magneto-optical effects which are claimed to be a function of a third magnetic field vector component that has been termed B(3). O of the central theorems of O(3) electrodynamics is the B-cyclic theorem:

B(1) x B(2) = i B(o) B(3)*, (93)

A conjugate product which relates three basic magnetic field components in vacuo defined as:

B(1) = �(�)√2 ( � � + �)exp(� ∅) (93a)

B(2) = B(0)√2 (-i i + j)exp(-i ∅) (93b)

B(3) = B(0)k, (93c)

where ∅ = ωt − ��, a phase factor, and i, j, k are the three unit vectors in the direction of the axes x, y, and z, respectively. Although the existence of the B(3) field has been a subject of controversy both pro and con over recent years, Evans recently claimed [96] that these magnetic field components encompassed by the relations (93a-c), along with the electric field components as well as the components of the magnetic vector potential (A), are themselves components of a Beltrami-Trkalian vector field relations (assuming the Coulomb gauge div A = 0). This is readily verified in the case of (93ab), since they present the form of the circularly-polarized solution to the Moses eigenfunctions of the curl operator we have discussed formerly in connection with turbulence in fluid dynamics.

Associated with the above developments, is the increasing importance given to hypercomplex formalisms for modeling the symmetries in elementary particle physics and quantum vacuum morphology. As discussed in former papers [97,98], the author believe that the most appropriate algebra for describing a hypothesized vortical structure for quantum-level singularities, as well as their macroscopic counterparts (Beltrami-type fields), is the biquaternion algebra (hypercomplex numbers of order 8) – the Clifford algebra of order 3, represented by the Pauli algebra Cl(3,0) such as previously examined in the Rodrigues/Vaz model. For instance, it is known that in a macroscopic Euclidean context, biquaternions are required to describe the kinematics/dynamics for the most general twisting (screw) movement of a rigid body in space [99,100]. It is therefore suggested that the most suitable formalism for screw-type EM fields of the Beltrami variety should transcend a traditional vectorial treatment, encompassing a para-vectorial hypercomplex formalism akin to the Clifford (Dirac) algebra used effectively to describe the electron spin in a relativistic context [101].

It is a conclusion in this regard that the founders of vector field analysis were remiss in failing to take into consideration account of the significance of the Beltrami field topology in addition to the traditional solenoidal, lamellar and complex-lamellar fields. An inclusion of the thorough examination of the Beltrami condition in the development of the vector calculus, would possibly have brought attention to the important intimate association of this field configuration with non-Abelian mathematical structure. If the history of vector analysis had taken this path, it is possible that the architects of vector field theory and classical electrodynamics, would not have been so quick to indiscriminately sever its connection from the natural quaternion-based foundation. Perhaps the recent work by Hillion/Quinnez, Rodrigues/Vaz, Evans, etc. [102,103] showing the necessity of considering non-Abelian models in electromagnetism, will be instrumental in helping to set the future of classical EM theory and vector field theory in general, on a firmer foundation.”

http://vixra.org/pdf/1207.0080v1.pdf

On 1/28/2014 3:38 AM, EMyrone wrote:

On 28/01/2014, at 7:00 PM, EMyrone wrote:

To Norman Page: I tried this link but I got "requested resource not available". Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, 't Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn't allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemis try, pe ace, literature, medicine and economics. What use is a non existent boson? From: norpag To: EMyrone Sent: 27/01/2014 18:11:09 GMT Standard Time Subj: Beltrami fields Forgot the link - Norman http://vixra.org/pdf/1207.0080v1.pdf

On 28/01/2014, at 7:00 PM, EMyrone wrote:

To Norman Page:

I tried this link but I got “requested resource not available”. Can you give me an idea of what it says about B(3)? Anything rude and out dated can be disgarded by now as a case of severe indigestion. The known animosity merhcants included Lakhtakia, Rodrigues, Bruhn, ‘t Hooft, Buckingham, Barron, Atkins and to a lesser extent Hehl (answered in comprehensive detail in UFT89). All of these have been disgarded entirely by the profession for several years now because they are so obviously biased and nasty. All criticisms of B(3) were answered years ago in all detail adn all these answers are well known. It was the other side who didn’t allow discussion, not me, as Gareth Evans pointed out immediately. So the Nobel Prize committee should get it right and get their facts straightened out. After the Higgs thing they are very nearly a laughing stock, and that is a pity. The Nobel Prize should be awarded for service to humankind as well as for the individual subjects: physics, chemistry, peace, literature, medicine and economics. What use is a non existent boson?