Electromagnetism using geometric algebra versus components

http://www.av8n.com/physics/maxwell-ga.htm

by fanf2 • 6 years ago

111 34 6 years ago

sgdpk• 6 years ago

Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the vector?

With GA, you associate the magnetic field with an oriented plane element. Oriented planes have this flip-mirroring property. Because space is 3D, you have exactly one vector to which this plane element is perpendicular, so you can identify the field with a vector. This is called the Hodge duality, and this particular form is accidental to 3D space.

If you need more convincing, the magnitude of the magnetic field is given by the cross-product, which calculates the area spanned by two vectors.

You could argue that you don't need GA for this, only to use the exterior product instead of the cross product. But by using an algebra that contains both vectors and oriented areas, you can effectively sum the electric and magnetic fields. This will give you pretty compact and beautiful equations, which is the essence of this article.

[1] https://en.wikipedia.org/wiki/Pseudovector

bollu• 6 years ago

For anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)

patrec• 6 years ago

For clicking convenience: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ (the link above is broken since HN includes the ]; in the URL).

pastrami_panda• 6 years ago

This is also an excellent and interactive blog-post building up to the understanding of how rotors can replace quaternions in GA: https://marctenbosch.com/quaternions/

aesthesia• 6 years ago

You still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.

yiyus4• 6 years ago

This "competition" between geometric algebra and differential forms makes me uncomfortable.

As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily done using the other one. What am I missing?

wwarner• 6 years ago

+1 to this exposition!

adamnemecek• 6 years ago

You should check out the bivector community https://bivector.net/ dedicated to Geometric algebra.

Join the discord https://discord.gg/vGY6pPk.

Check out a demo https://observablehq.com/@enkimute/animated-orbits

DreamScatter4• 6 years ago

Recently I made a video presentation about geometric algebra, including the formulation of Maxwell's equations in GA: https://youtu.be/7hlDRLEhc8o

Also, my implementation of geometric algebra in the Julia language, Grassmann.jl https://github.com/chakravala/Grassmann.jl

It helped me truly understand Maxwell's equations for the first time, understanding that it is not just some physical artifact but actually a natural foundational idea in pure mathematics applicable to physics.

taliesinb• 6 years ago

That’s cool.

What do you make of the algebra of the dihedrons? https://youtu.be/lqH4BLHGsFw . It’s a “sister algebra” of the quaternions.

jacobolus• 6 years ago

These “dihedrons” (which you might as well just call “2x2 matrices with real entries”) are isomorphic to the geometric algebra of the 2-dimensional Euclidean vector plane with signature (+, +), with 1 scalar component, 2 vector components, and 1 bivector component. Personally I find the basis 1, e₁, e₂, e₁e₂ and the notation of GA to be conceptually much clearer than the matrix entries and standard matrix sum/product to work with, but sometimes using a matrix representation is convenient in a computer.

(2x2 real matrices are also isomorphic to the geometric algebra of the 2-dimensional pseudo-Euclidean vector plane with signature (+, -), under a different interpretation.)

By comparison the quaternions are the even subalgebra of the geometric algebra of 3-dimensional Euclidean vector space with signature (+,+,+), consisting of only the 1 scalar and 3 bivector components. Or under a different interpretation are isomorphic to the full geometric algebra of the 2-dimensional vector plane with signature (-, -). They can be represented as Pauli matrices.

For more on this see the papers and books of Garret Sobczyk, https://garretstar.com/secciones/publications/publications.h... ; for example the recent https://www.garretstar.com/sobczyk09-mar-2020.pdf

enriquto2• 6 years ago

Nobody ever writes Maxell's equations using "components", the title sounds like a straw man argument.

The rest of the text is well written, but hopelessly useless without a comparison with the typical way to write Maxwell's equations using differential forms (which turns out to be essentially identical to geometric algebra).

AnimalMuppet• 6 years ago

If I understand correctly, Maxwell wrote Maxwell's equations using components. That's why they took 20 equations.

enriquto2• 6 years ago

Sure. My first sentence is missing the word "today".

peter303• 6 years ago

Maxwell wrote his equations using components. Heavyside create modern vector notation.

bawana3• 6 years ago

If a point is the intersection of two lines, and a line is the intersection of two planes, then what structures intersect to give a plane? Two 3D spaces? And so the intersection of 2 4D spaces should give a 3D space? How do I express these ideas with GA?

jbay8082• 6 years ago

Total non expert here but your fascinating question got me thinking out loud.

Two 3D volumes in a 3D space obviously intersect to give another 3D volume, unless they're tangent. So at first glance I'd be tempted to say no, but...

Two overlapping polygons in a 2D space also define another 2D polygon. Two overlapping line segments in a 1D space define another line segment. You only get a reduced-dimension object at the intersection if they're intersecting in a higher dimensional space. Otherwise they can't intersect "at an angle".

So I'd say purely by following the pattern that it must be that two 3D spaces intersecting "at an angle" within a 4D space define a plane by their intersection.

bawana3• 6 years ago

That's why I ask how to formally describe this problem. It's too 'squishy' to say two 3d spaces intersecting at an angle in a 4d space. I thought perhaps I could use time to untangle this visualization-defining a solid as the result of taking a plane and sweeping it through space. So, envisioning the 'growth' of a solid from a moving plane and imagining the intersection of the growing solid with the another growing solid from a different plane.

wnoise2• 6 years ago

A point is the intersection of two "generic" lines in an ambient 2-d space. But you can have coincident lines which intersect in a line, or parallel non-coincident lines which have no intersection. Further, in 3-d, the majority of lines are "skew", and neither parallel nor intersecting.

Similarly in an ambient 3-d space, 2 generic planes intersect to give a line. But the planes can also be coincident or parallel. And in higher ambient spaces can intersect at only a point, rather than a line, or even fail to intersect in a non-parallel way.

In 4-d space, the intersection of two generic 3-d spaces does indeed give a plane, with exactly similar caveats.

The standard GA doesn't directly represent general lines or planes, however. The elements are the equivalent of "vectors" rather than "points", and always go through 0. The obvious way to handle these are parameterizing the lines and surfaces, but you're essentially working with equations for the surfaces, and keeping track of the variables.

The slick way of handling it is with _projective_ geometric algebra, and intersections turn into "meets". The meet of two parallel lines (planes) is now a "point (line) at infinity", and of a (line, plane) with itself is the line (plane) again. Skew lines have a meet of 0 (not the point 0, the number 0).

bawana3• 6 years ago

thanks again internet. some unknown intelligence, ironically named, wnoise, is confirming my intuition. That a plane is the result of two intersecting 3d volumes in a 4d space. And 'our' 3d space (where the meat lives) is the intersection of 2 4d volumes in a 5d space. But dear Wnoise, enlighten me about ' _projective_ geometric algebra, and intersections turn into "meets".' Where can I find the book, 'Projective_ geometric algebra' for dummies?

carapace• 6 years ago

I found "Geometric Algebra for Electrical and Electronic Engineers" helpful: https://ieeexplore.ieee.org/document/6876131

ABSTRACT: In this paper, we explicate the suggested benefits of Clifford’s geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.

Electromagnetism using geometric algebra versus components

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