Broken Maxwell EM ?

[EricLux]

Wow you're doing better than me! You find Elements after I have looked for so long!!
I found most of my insights into the EM side out of Maxwell's "treatise." He gives a few
formulations, and mention of their use. Also Joly and Tait. P.G. Tait's introduction has a very
fine low level trigonometric approach, where he concentrates on parameterization of a line.
Also Joly became impressed with the elegant and concise notation possible, and I note on
from a personal point of view, that several points made by Maxwell appear to use Joly-ish
style notation.

Hamilton was a mathematician primarily, and proposed his math be used. Sorry if I gave you
the impression that he derived EM. Hamilton did not. Maxwell is the only one I know of who used it
rigidly for classical EM, though others have touched on it. JCM lists a number of citations in the "treatise" index, but the body of the work uses quaternion notation frequently. As a motivator, examination of JCM's discussion of periphractic regions (early in vol1) one will find the need for vector spaces that can be adjoined and treated rigorously as a graph with vector products that correctly couple across the membranes. If I had the "treatise" in front of me (it's at home), I'd cite some of my favorite lines. Maybe I can pull something together for later this week.

When treating problems of statics and dynamics using this algebra, one finds that the remaining term is the total force pushing inwards on the system (the convergence). Problems we are trying to solve currently, Poynting flow vis-a-vis multiple axis E&B sensors, spectrum of local curvature fluctuations given plane waves, decomposition of vector field measurements using the two-scalar solution of Whittaker (see: 1903), 4D DFT.

toybot

That sheds some more light on me, thank you!

 

[EricLux]

Ark,

You said that you don't see anything special with the quaternion algebras of which
I agree by itself but perhaps used in the context as Maxwell did, perhaps there is
something else to consider here?

There was a "big fight" regarding Maxwell's use of quaternions in his "A
treatise on Electricity and Magnetism", 1873 release (1st release) as started
by Heavyside, Gibbs, and also opposed by Lord Kelvin (?).

From what I have read - some are saying that the vector form has diluted some
of the properties of the 20 equations of 20 unknowns into a "neat" 4 vector equations
by use of arbitrary forced guage conditions? I also read that quaternions are best used
when working with spatial/rotational bodies in space and it being used today for solving
issues regarding satelites which are more difficult and error prone in other forms. A link
is provided. Please understand that I am a novice so I do not have a full background of
the all the technical stuff to be in the know, so I am learning as I go.

Here are the following claims or statements by others regarding the
Heavyside-Gibbs v.s. Maxwells quaternion based equations regarding
EM theory:

[links of no particular order]

hypercomplex.com

This domain may be for sale!

www.hypercomplex.com

http://homepages.ihug.co.nz/%7Esai/Beard_scal_vac.html

http://www.zpenergy.com/downloads/Orig_maxwell_equations.pdf

http://www.cheniere.org/correspondence/030706.htm

links: [google search: quaternion rotational dynamic]

http://www.absoluteastronomy.com/enc3/quaternions_and_spatial_rotation

The last link in particular had me interested because it seemed to me that maybe
there is a sychroncity (or pattern) that since EM travels in spiral/helix/rotation path,
then perhaps the quaternion form is the best form to use rather than that of vectors
and will result in no loss of [or hidden] properties?

Do you believe that Maxwell's quaternion based forms are fully/accurately represented
today by use of the Heavyside/Gibbs (vector notiations) form used by many EE's today
and that nothing was "lost in translation"?

Kind regards,
Dan

Thanks Dan

 

[EricLux]

I essentially finished my inquiry into the subject. The result is: there is nothing mysterious whatsoever about the use of quaternions by Maxwell. They are used in the edition of 1873 as well as in the third edition, reprinted by Dover. Maxwell is using quaternions because at that time the vector notation was not yet developed (the operators grad, curl and div).


It is true that Maxwell was changing his equations (both in their quaternionic and component form) with time. They are somewhat different in the first edition and in the third edition

, and still different in Maxwell's "Dynamical theory of the electromagnetic field" published in 1864. Today we group
the quantities differently. There is some discussion of these issues in the Russian edition of Maxwell's "Selected works" where the Editor and Translator compare Maxwell's original equations with what we call Maxwell's equation today. Here is the relevant part:

The main difference from the contemporary formulation is that Maxwell includes the term -[Bu] in his definition of E - eq. (D) - while today we include this term in the Lorentz force, not in E

Some of those disinformation spreading people say: Maxwell had 20 equations, much more than we have today ! This is a joke. Suppose I have
one equation:

X= 2A

Now I introduce new variable B=4A

Then I have TWO equations instead of one:

X=B/2
B=4A

By defining new variables we introduce new equations. So, the number of equations is not relevant. The number of "essential equations" is important. And here there is no mystery whatsoever.

Summarizing: all this internet gossip about mysterious quaternions and Maxwell is a disinformation, it is not not based on the data. The sites spreading rumours about mysterious quaternions can be added to the list of disinfo sites.

The discussion of Maxwell equations today continues. Here we have, for instance, a paper by two Russian: Andrew E. Chubykalo and Vladimir V. Onoochin, now in Mexico: On the theoretical possibility of the electromagnetic scalar potential wave spreading with an arbitrary velocity in vacuum. In the introduction we can read:


The paper was sent to Phys. Rev. letters, and was rejected. Mainstream journals do not accept
papers that question mainstream paradigm! This is a well known fact. So the authors have to
publish original papers in fringe journals. Sometimes really really fringe! To distinguish
a "nonsens" from "original and new" is not an easy task and only experts can do it - sometimes experts
make mistakes as well.

Thanks for the insight and the various links.