New foundations for classical mechanics

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1. jordibcJan 2023
   In case it helps:

   In addition to "Geometric Algebra for Physicists" (whose first two chapters I'd recommend to get a nice overview), I found Hestenes' "New Foundations for Classical Mechanics" to be very good and readable.

   Also, there are many good resources in https://bivector.net/ , including videos, papers, presentations and programs.

   Finally, an interesting paper (that got me kickstarted) is "Imaginary Numbers Are Not Real—The Geometric Algebra of Spacetime" by Gull, Lasenby and Doran.

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2. jacobolusFeb 2022
   What I have repeatedly found with GA is that I can solve some problem I have using some other brute-forceish tools with a few pages of tricky error-prone scratch-work that balloons out to a complicated mess before simplifying back down at the end, and then afterward think about it a bit and come up with 2–6 lines of simple GA identities showing the same thing in a much higher-level coordinate-free way, and with most of the steps geometrically interpretable, rather than just opaque calculation. But coming up with the simple version at the beginning is hard.

   The tricky part about it is that there are a lot of useful identities that can be written down, and properly learning a decent number of them and figuring out which ones to apply in which situation takes probably years practice, ideally with some guidance/support from someone who knows more than you. (I do not feel like I have mastered the subject.) The same thing happens using whatever other formalism, with the difference that many identities that are pretty short to write down in GA are much more complicated to write down, so people don’t even try to use them.

   I’m not sure if there’s really a good beginner source, but I haven’t ever really sat down and tried to go comprehensively through the exercises in any books pitched at a relatively elementary level. You could try Alan MacDonald’s book Linear and Geometric Algebra which is designed as an intro undergraduate textbook. If you want to also learn some mechanics, you could try Hestenes’s book New Foundations for Classical Mechanics.

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3. jacobolusNov 2021
   > Suppose I want to teach first-semester mechanics

   If you need to teach undergraduate mechanics, I highly recommend you at least read some of Hestenes’ New Foundations for Classical Mechanics http://geocalc.clas.asu.edu/html/NFCM.html

   > without any distinctive advantages

   The most basic distinctive advantage is that you can invert vectors (which is incredibly useful!!) without needing to pretend that vectors are matrices, complex numbers, or some other kind of object.

   GA takes most of the advantages of complex numbers vs. R² for representing plane geometry, but extends them to arbitrary dimension, and extends them further (when using complex numbers for plane geometry you end up representing vector–vector products via the obscure z̄w product involving complex conjugation, and it is easy to get confused about the difference between a vector vs. a scalar+bivector).

   But there are a wide variety of other powerful (and geometrically interpretable) algebraic identities which can be applied to vectors, blades, and multivectors, ranging from awkward to impossible to express using the language of differential forms, Gibbs-style vectors, etc. Physicists often end up resorting to tedious coordinate-by-coordinate calculations for stuff that would end up being an easy vector expression in GA. Learning these identities and how to apply them takes years and a lot of practice solving problems using GA.

   My own experience for the first few years of knowing that GA existed but not being too fluent with it was that I would work some problem (mostly 2–3 dimensional geometry problems) out in coordinates, spending like 2 pages of scratch paper for the opaque intermediate calculations, with high chance for mistakes, then eventually find that most of the ugly bits along the way canceled and yielded a nice result. Then I would think a bit more about the problem, skim through a list of GA identities, and find I could have shortened that 2 pages of work to 3 lines, each of which had an obvious geometric interpretation.

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4. SAI_PeregrinusFeb 2020
   In addition to gibsonf1's recommendations (not books), the following books are good. The first 2 are more pure introductions to the math, the third is applying it to physics, the fourth to CS.

   Linear and Geometric Algebra, by Alan Macdonald: http://www.faculty.luther.edu/~macdonal/laga/

   Vector and Geometric Calculus, by Alan Macdonald: http://www.faculty.luther.edu/~macdonal/vagc/index.html

   Application to physics: New Foundations for Classical Mechanics by David Hestenes: https://books.google.com/books/about/New_Foundations_for_Cla...

   Geometric Algebra For Computer Science by Dorst, Fontijne, and Mann: http://www.geometricalgebra.net/index.html

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5. jacobolusApr 2019
   What is your background/perspective? For introductory motivation, check out https://www.shapeoperator.com/2016/12/12/sunset-geometry/

   And perhaps follow up with the GA for Computer Science book, or if you like (or want to learn about) Newtonian mechanics, try Hestenes’s book New Foundations for Classical Mechanics.

   For people with a mathier background, I would recommend https://arxiv.org/abs/1205.5935

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6. eggyDec 2017
   I started down the GA path in 2005 with the purchase soon after of Geometric Algebra for Physicists, and other than learning the simplicity of expressing things in this algebra, I have not been able to dive deep into its uses. I have checked out Versor[1] for code stuff, and there was the promise of potential hardware implementations of the algorithms necessary to be able to make gains in the human program GA code to the pipelines and bits on CPUs, GPUs and other hardware.

   I think the day is coming for an explosion of GA in everything, since it has been simmering behind the scenes for many years thanks to many, Dave Hestenes[2] in particular. I love his book New Foundations for Classical Mechanics.

   I program in APL and J[3], and hope that the array programming will segue into a winning GA library/module and way of approaching coding GA to make it more of a player in my work rather than my hobby.

   EDIT: Versor.js is also available for a spin! [4]

     [1]  http://versor.mat.ucsb.edu/
     [2]  http://geocalc.clas.asu.edu/
     [3]  jsoftware.com
     [4]  https://github.com/weshoke/versor.js/

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