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Why is Maxwell's theory so hard to understand? (2007) [pdf]

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Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#71

Post by jbay808 »

tntn wrote: post #12
madengr wrote: post #11

Earlier quoted context omitted.

Maxwell unified electromagnetism into a propagating wave. That was the big leap. Before that, things were disparate. Hertz proved it experimentally, and Heaviside turned it into an engineering discipline.

This is true of the mathematics, but it's worth noting that Faraday had already proposed EM waves ~30 years earlier based on his intuitive understanding.

And had rejected the aether, on the basis that it was unimaginable that any medium could propagate transverse waves in all directions but not allow longitudinal waves.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#72

Post by jle »

aj7 wrote: post #48
nonbel wrote: post #16

Earlier quoted context omitted.

>"E&M is far from "simple". It contains special relativity, for starters." A theory (which is just a set of assumed first principles and rules of logic) can be simple but allow you to deduce vast complexity from it. In fact, it is ideal for a theory to be as simple as possible. The "game of life" is not really a theory, but it demonstrates that simple rules can lead to surprising complexity: https://en.wikipedia.org/…

I reflexively downvote the game of life. I have never heard of a single useful application or analogy of “finite automata.” Yes, there are speculations about extremely small Planck lengths, by no new physics has panned out. Finally I am sad that one of the greatest computer scientists and polymaths, Ed Fredkin, got sucked into this. We all have weaknesses.

GoL wasn't used here because of any value or analogy... it's used as an example of how a simple set of rules can generate complex behavior. GoL can display complex behavior, but it would be a stretch to to say that the rules governing it are complex. If you think GoL is too simplistic, you are agreeing with the original comment.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#73

Post by killjoywashere »

amatthew wrote: post #64
killjoywashere wrote: post #57

It's not until now, over 20 years later, that I realize what a magician my E&M prof was. After months of careful development of basic E&M, Maxwell's equations simply fell out. As I recall, it was capacitance that was the impetus. It was as though we had struggled in a dark tunnel for months and suddenly came around a turn and there was light. Not a little, but sheets of daylight and all the equations just fell togeth…

In what ways were they that important later in life?

Because my undergrad is in physics and I'm still in science, E&M opportunities are evident to me, but I know I would struggle to really go after them myself. So I get help, but I sure wish I could do it on my own.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#74

Post by GregBuchholz »

tim333 wrote: post #19
k9s9 wrote: post #2

Interesting. And makes me wonder what else we are missing and misunderstanding, as we waste so much time trying to express the complexity of the world in spoken language rather than math. Not just physics but in sociology, psychology, economics etc. Faraday purely from experiments 30-40 years before Maxwell, intuitively understood electromagnetism. Nobody serious believed it because he didn't have the training to exp…

Spoken language has a lot going for it. I note the featured article and it's discussion are all words not maths. By the way looking at Maxwell's paper it seems awful complicated https://royalsocietypublishing.org/doi/pdf/10.1098/rstl.1865... compared to the modern form of the equations https://ethw.org/w/images/d/d6/Maxwell_image_02.jpg

>compared to the modern form of the equations

Are the Geometric Algebra...

    ∇F = μ₀cJ
..and Differential Form...

    dF = 0
    d⋆F = J
...equivalents post-modern?

https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#75

Post by jacobolus »

iorrus wrote: post #36
ncmncm wrote: post #21

It is unfortunate that Dyson neglects the role of Oliver Heaviside, again. This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner. Heaviside invented the mathematical tools we still use to understand and teach Maxwell, and most of the important consequences of the theory, but Pupin, Hertz, Marconi, and deForest used his methods and took got the credit. Today Heaviside…

Exactly, one of the main reasons why maxwell is so hard to understand is that everything is expressed using quarternions unlike Heaviside who expressed the equations using the vector notation we see them expressed in today. In reality ‘Maxwell’s’ equations are in fact Heaviside’s.

Arguably the reason that generations of STEM students have been horribly confused about 3-dimensional vectors and rotations (including electric/magnetic fields), etc. is that they were reframed in the confused and non-generalizable Gibbs/Heaviside language, instead of in Grassmann/Clifford’s formalism in which vectors and bivectors can be properly described as separate types of objects.

It can be so much nicer. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#76

Post by DoctorOetker »

jules wrote: post #61
wmnwmn wrote: post #14

E&M is far from "simple". It contains special relativity, for starters. Also it is incompatible with thermodynamics: solving this problem is why Planck invented quantum mechanics. Also point charges have infinite energy: this problem leads to renormalization theory. Also it introduces gauge invariance, an essential but complex part of all modern theories. And lastly, the mathematics of E&M is a big step up from Newto…

E&M is simple in the sense that the equations that govern the relationship between the charge distribution and the electromagnetic field are quite simple. And the equations governing how charges move in an electromagnetic field are quite simple. I think this is what Dyson has in mind. Understanding the implications and limitations of those equations is not simple at all. An even more blatant difficulty with E&M, rela…

with schizophrenic you mean a system of coupled differential equations? in that case I wouldn't agree with schizophrenic.

or are you referring to the fact that mathematically speaking the system of differential equations is indeterministic due to mathematically perfectly valid advanced potentials? in this case I agree that there may be much more to learn from the Maxwell equations, and I have no qualms with naming this feature (highly) schizophrenic.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#77

Post by jacobolus »

tntn wrote: post #41
srean wrote: post #35

Earlier quoted context omitted.

> Quaternions are not a "model of vectors. I dont think I agree. Yes they are not the Gibbsian vector but they are a pretty damn good model of what we want to model as vectors. Its already a little closer to geometric algebra / calculus than Gibbsian vectors. Its a pity that Grassman and Clifford's work gained attention much later. Once javascript is out of the bag one cant do much about it. Thats how I feel about Gi…

I think geometric algebra is a very cool and elegant approach as well, but it has not yet proved useful or valuable to working engineers. If we want to compare to programming languages, perhaps Haskell vs. C. One is beautiful and elegant, the other is down to earth and conceptually simple for "normal" people. One is primarily used by academics and researchers, the other is used to build the world. Both are admirable.

> not yet proved useful or valuable to working engineers

The people who value it are those who must deal concretely with the details of the model on a daily basis: computer programmers working in graphics, computer vision, robotics, physical simulation, ...

It doesn’t do all that much for many pure mathematicians who are happy to hand-wave away the fine details content with the knowledge that the languages can be proven equivalent by someone else, just like it doesn’t do much for the type of engineers who just use someone else’s software for their work.

Of course, the current mess is a major point of pain/confusion for the generations of undergraduates studying vector calculus. But they should go through the same hazing ritual the rest of us did, right? It’s only fair.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#78

Post by jacobolus »

earthicus wrote: post #69
jules wrote: post #67

Earlier quoted context omitted.

Yes, they operate on the same objects. The main operations on differential forms are the exterior derivative d and the wedge product f /\ g. These are independent of the metric. If you add a metric you get one extra operation called the Hodge star ⋆, which is a prefix operator, so it operates as ⋆f. Alternatively, if you have a metric on your space you can define the geometric algebra derivative D and the geometric p…

Is it possible to work with geometric algebra effectively without a metric using some more complicated construction?

Sure. What model to use depends on you are trying to model.

http://geocalc.clas.asu.edu/pdf-preAdobe8/PGwithCA.pdf

http://geocalc.clas.asu.edu/pdf-preAdobe8/DLAandG.pdf

etc.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#79

Post by DoctorOetker »

iorrus wrote: post #36
ncmncm wrote: post #21

It is unfortunate that Dyson neglects the role of Oliver Heaviside, again. This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner. Heaviside invented the mathematical tools we still use to understand and teach Maxwell, and most of the important consequences of the theory, but Pupin, Hertz, Marconi, and deForest used his methods and took got the credit. Today Heaviside…

Exactly, one of the main reasons why maxwell is so hard to understand is that everything is expressed using quarternions unlike Heaviside who expressed the equations using the vector notation we see them expressed in today. In reality ‘Maxwell’s’ equations are in fact Heaviside’s.

the physics equivalent to vi/emacs flamewars doesn't need any more fuel please...

https://en.wikipedia.org/wiki/Nabla_symbol#History

Also curious, which of Maxwell's texts you are referencing?

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#80

Post by wnoise »

iorrus wrote: post #36
ncmncm wrote: post #21

It is unfortunate that Dyson neglects the role of Oliver Heaviside, again. This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner. Heaviside invented the mathematical tools we still use to understand and teach Maxwell, and most of the important consequences of the theory, but Pupin, Hertz, Marconi, and deForest used his methods and took got the credit. Today Heaviside…

Exactly, one of the main reasons why maxwell is so hard to understand is that everything is expressed using quarternions unlike Heaviside who expressed the equations using the vector notation we see them expressed in today. In reality ‘Maxwell’s’ equations are in fact Heaviside’s.

It was my understanding that the original Maxwell equations used separate letters for each component rather than indices or any modern method of writing down the components as a unified thing (so the four expand to at least 12 separate equations (20, once you factor in constitutive and continuity equations)). Later on, he did use quaternions to organize like quantities, as we would use vectors, but still generally worked component-by-component. But yes, the most common modern versions are effectively due to Heaviside.
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