Okay, the introductory quaternions post went out last Monday. This weekend I expect to write about all four representations of three-dimensional rotations: rotation matrices, angle & axis of rotation, quaternions, and Euler angle decompositions.
I wasn’t sure last weekend what I would do besides quaternions… it turned out to be a control systems simulation. Nothing fancy… okay, it was a fairly complicated “plant” being hit with a rectangular wave (bang-bang). I was just trying to duplicate something in a book. The challenge was that the book (Ellis) expected me to be using its companion software on a pre-defined “experiment”… and did not provide all the parameters for an independent simulation.
It was nice to see that I could do it with Mathematica®.
Nevertheless, my mathematical life appears to have gotten more complicated.
I was chatting with a physicist friend of mine, and I told him that the Yang-Mills equations that give us quarks are equivalent to the study of connections on fiber bundles. What I probably should have said was (from my own post of Feb 28, 2009):
“It was not until the early 1970s, however, that dawn broke and,
in the clear light of day, it was recognized that a gauge field
in the sense of the physicists is essentially nothing other than
the curvature of a connection on some fiber bundle.”
(p. vii, Naber, Gregory L.; Topology, Geometry, and Gauge Fields: Foundations. Springer,1997. ISBN 0 387 94946 1.)
And from July 11, 2009: … fiber bundles. They generalize the tangent bundle, which is the set of all tangent spaces to a manifold. What you should visualize is a circle with a whole lot of tangent lines drawn on it. And if you draw enough of them, you can dispense with the circle itself, which suggests that we can study a manifold by studying associated fiber bundles.
But I couldn’t do much more than talk about a cylinder and a Mobius strip, and the tangent bundle, as examples of fiber bundles. And my conversation was limited as much by my lack of mathematics as by his.
Now, once again, I feel an overwhelming urge to curl up with my graduate differential geometry texts and study fiber bundles. (It turns out that the text for the graduate course I took did in fact mention bundles for about 4 pages… but the professor wasn’t actually using that book at all, so I never saw them.)
I know perfectly well, however, that this will take a while. In fact, it will take way longer than I want it to… but there’s not much help for that: it will take however long it takes. Realistically, I’ll probably play with them for a while, learn something, and put them down….
And in the meantime I will keep working on Boolean equations, and elementary classical control theory, and everything else that I get distracted by. And, as you may have noticed, I am easily distracted by applications of mathematics.
So don’t despair… I’m not likely to get very far from applied mathematics, and not for long even when I do.
Now what? Rotations… fiber bundles… elementary control theory… Boolean equations…. I don’t know which one to pick up first.