Comment by enkimute | Hacker News Reader

Comment by enkimute | Hacker News Reader

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Not the same concrete example, but one where I do find the Geometric Algebra version substantially more insightful, is the treatment of rigid body mechanics in the geometric algebra of the Euclidean group (R_{n,0,1}).

It has the dual quaternions as even subalgebra (in 3D), and unifies all linear and angular aspects. It leads to remarkable new insights, as removing the need for force-couples (pure angular acceleration is caused by pushing along a line at infinity), while pure linear acceleration is caused by forces along lines through the center of mass.

These geometric ideas are independent of dimension - forces, both angular and linear are always lines. The treatment of inertia becomes a duality map, and things like Steiners theorem are not needed at all.

On top of this, the separation of the metric that sets GA apart means that this formulation of rigid body dynamics works not only in flat Euclidean space, but unmodified in the Spherical and Hyperbolic geometries. (by a simple change of metric of the projective dimension).

For a (graphics/game programmer oriented) tutorial on this see https://www.youtube.com/watch?v=pq9YfdPHhIo&ab_channel=Bivec...

Well, I think the point is that in rigid body dynamics, the configuration and phase spaces naturally form a manifold and then the equations of motion are in terms of differential forms on the cotangent bundle of the these manifolds. This is commonly expressed in terms of the language of exterior algebras, hodge duals, etc. That's what is driving all of this, and is usually covered in a good class on mathematical physics. Again, there is nothing new here except marketing, but marketing plays an important and useful role.

I remember for a long time, people coming from the math end of things would look down a bit on physicists laboriously working everything out in complex tensor notation when there are these elegant canonical descriptions arising from differential geometry that look very simple and beautiful and are completely coordinate-invariant.

But then when you want to actually calculate something, you end up doing all the painful tensor contractions anyway, so the physicists would likewise often lookdown on the mathematicians for writing these simple one liners that described all of mechanics but not really understanding how to calculate stuff.

So if repackaging some of the basic facts of differential geometry as "Geometric Algebra" gets physicists to be excited about it, then that's a good thing. Just like repackaging some of the laborious tensor calculus computations into differential geometry has gotten a lot of mathematicians excited about physics. It really is much more pleasant to work in a coordinate-free manner using differential structures associated to the natural manifold suggested by the problem, rather than being stuck in euclidean space and needing to deal with lots of fictional forces and complex change of basis formulas.

For example, look at this text: https://depositonce.tu-berlin.de/bitstream/11303/2482/1/Doku...

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