Happenings – 21 June

I’m working on several things, and it’s possible I’ll have a technical post ready later today. OTOH, I’m going to a dinner party tonight, so this will be a short schoolday.

As you might guess from the recent posts about rotations, I have gotten caught up in rotating coordinate systems. The original cause was a nifty equation in the airplane control books. As is true of too many things, I can even find that equation in an old schoolbook, in this case my ancient copy of Goldstein. Worse, I highlighted it all those years ago. That equation writes the rotation axis \omega in terms of the derivatives of the Euler angles and their rotation axes.

Right now, however, I am playing with two more familiar and elementary equations. Well, you would have seen them in upper-division classical mechanics; before that, you usually just get them in fragments (radial and tangential components). Eventually they are usually written something like

v_f = v_r + \omega\ \times\ r

a_f = a_r + 2 \omega\ \times\ v_r + \omega\ \times \left(\omega\ \times\ r \right)

(If you recognize them at all, you may know the 2 \omega\ \times \ v_r term as the Coriolis effect.) I have a derivation of them that gives me some insight into exactly what’s what. It’s at what I call stage 2: the math is all written out, but I need to add the surrounding discussion.

Playing with these equations has sometimes interfered with my making regular progress in the study of surfaces (Bloch). Oh, I still need to address an old comment about gluing schemes….

A friend has picked up the Cox-Little-O’Shea book on algebraic geometry, so I’m working thru that with him.

And, of course, there’s principal components / factor analysis. I’ve done most of the remaining work for Malinowski, but I need to add a lot of commentary, too. And I have one sort-of-open question about something he did.


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