If you haven’t done so, check out ‘Symmetry and Separation of Variables’ by W Miller, Jr., Addison Wesley, 1977 .

It’s good to see the blogosphere lit up by an enthusiastic discussion of a topic as old school as spherical harmonics. There is something rather satisfying about that stuff; even our host seems to get a thrill out of it.

As for the blast from the past phenomenon - the spotting and exploitation of analogies is a big part of physics research; this episode could be a case study in the way it is done. I remember solving a problem in wave propagation in a stratified medium by retrieving and brushing up a half-forgotten exercise in Landau and Lifshitz Quantum Mechanics. Is that an examinable skill? I fear that, back in the day when both examiners and candidates were real men, it would have been regarded as bad form, if not outright cheating. A referee shared this view, demanding that any mention of matters quantum mechanical be excised from a hydro paper.

BTW Whitaker and Watson’s ‘Modern Analysis’ is the real deal when it comes to special functions Chapter 18 dispatches ‘the equations of mathematical physics’ in less than 20 pages, and invents twistor theory in the process.

Oh, and I second Anon’s recommendation of Gradshteyn and Ryzhik. Every theorist should have a copy.

The fact that the Laplacian is seperable in many coordinate systems, including ellipsoidal ones, has always fascinated me. I presume that perturbations along some direction turns the initial spherical space into an ellipsoid. Did you get to use this result in any form?

functions.wolfram.com

It’s like a searchable, more extensive, A&S.

Link

For example, if space is hyperspherical (which maybe it isn’t due
to current expansion, but in principle it could have been) then consider
what happens inside a box “actually moving” through space: since internal test bodies follow great circle routes (geodesics) on the hypersphere, they will oscillate back and forth inside the container as it travels around the hypersphere at a rate proportional in part to “velocity” relative to the geometric construct itself. (Consider that their individual geodesics can’t stay parallel; they must intersect like longitude circles on the
earth.)

This is simply unavoidable, and means that once you have rules about
“following” this or that path on a curved surface, then it becomes a
standard for referencing motion on that surface. Well, relative to what? I
would suppose, the average motion of matter, but this issue really isn’t
adequately clarified, and not adequately presented in discussion and
textbooks. (Sure, Taylor and Wheeler show ships coming together while
traveling the surface of the earth in _Spacetime Physics_, but don’t really
apply that to the implications for standard of motion in the universe.)

… the environment inside the moving box really is a special frame, because objects experience a tidal field with no internal sources, or specific external ones (it’s not like passing by a planet….) because this is an effect from the overall density of the universe … small test particles spontaneously fall towards and away from the center of the box even though there is no mass density *within the box* (well, I assume we can keep other matter out of the box as it moves, and still maintain sufficiently geodesic motion for the test particles.) But yes, OTOH, particles falling to the center of the earth do that also, so the inside of the box is a sort of tidal field. The important thing to me is not whether the box environment should be called “special”, but the very fact we can tell how fast we are “moving through the universe” - IOW, the preferred frame for velocity, not “special” in any sense beyond that. See the interesting comments by Ben Rudiak-Gould [of UC Berkeley] on this
issue [in this sci.physics.relativity thread.] (In particular, it isn’t
what can be simulated by acceleration or rotation in an inertial space.)

Note, this goes beyond physics, because it means we can define “motion” on a featureless geometric body once we have rules like the following of a geodesic. That is counterintuitive given there not being “marks” on the hypersphere to reference relative motion ….we can indeed tell that we are moving on the featureless surface in this case, that’s the whole point of my illustration. (This post has two points, the first is about motion in a real physical universe - even if ours isn’t curved that particular way. The second point is about motion “on a mathematical sphere”, where oddly we can also tell “how fast we are moving”!
We can tell how fast we are moving by the rate of acceleration of the test bodies inside the box. We aren’t moving relative to the frame except to imagine that something of the surface is absolute as a holder of rest, since we aren’t moving farther or closer to anything (uniform curvature
everywhere, and we stay inside the manifold.) This *is* odd and
counterintuitive, since a “mathematical entity” like a sphere isn’t suppose
to be thought of the way we’d consider a steel ball, that we know how fast we are sliding along because there’s a “real surface” to be moving past constituent particles of. I think it shows something about “time” being special as a concept.