Various toroidal coordinates used by M&F
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A short working note by Phil dated 12.30.13 that surveys which toroidal coordinates he used in his Stackel paper and his Bowl paper, and which Morse and Feshbach (M&F) use on page 666 of Volume 1 and page 1301 of Volume II. It lists scale factors, the Cartesian coordinate formulas, and the separated harmonics built from Legendre functions. It concludes that the (μ,θ,φ) and (μ,η,φ) sets of M&F match his (ξ,u,φ), while M&F's ξi set on page 666 is a different coordinate system.
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Various toroidal coordinates used by M&F PhL 12.30.13
1. What toroidal coordinates do I use in my Stackel paper?
Answer: I use (ξ1,ξ2,ξ3) = (η,θ,ψ) where the latter are the M&S coordinates! These are scale factors:
h1 = h2 = a/[ch(ξ1)-cos(ξ2)]
h3 = a sh(ξ1)/[ch(ξ1)-cos(ξ2)]
I then compute a Stackel matrix using these coordinates:
Φ = = (7.7)
Later I get these same matrices doing the calculation a different way.
I get these three ODE's
L1X1 = ∂12X1 + coth(ξ1) ∂1X1} + [(1/4) - k22 - k32/sh2(ξ1)]X1 = 0
L2X2 = ∂22X2 + k22X2 = 0
L3X3 = ∂32X3 + k32X3 = 0 .
and then I get the harmonics two different ways:
osc osc
[Pqp-1/2(chξ1), Qqp-1/2(chξ1) [ sin(pξ2), cos(pξ2) ] [ sin(qξ3), cos(qξ3) ] * [ch(ξ1)-cos(ξ2)]1/2 .
osc osc
[Pqiτ-1/2(chξ1), Qqiτ-1/2(chξ1) [ sh(τξ2), ch(τξ2) ] [ sin(qξ3), cos(qξ3) ] * [ch(ξ1)-cos(ξ2)]1/2
2. What toroidal coordinates do I use in my Bowl paper?
The first thing I do is just state these two atomic forms:
ξ in (0,∞) u in (0,2π) φ in (0,2π)
expo osc osc
(1) [Pn-1/2m(chξ), Qn-1/2m(chξ) ] [ sin(nu),cos(nu)] [ sin(mφ),cos(mφ)]
osc expo osc
(2) [Piτ-1/2m(chξ), Qiτ-1/2m(chξ) ] [exp(τu), exp(-τu) ] [ sin(mφ),cos(mφ)]
So here is the connection to the Stackel paper
Stackel paper (ξ1,ξ2,ξ3) = (η,θ,ψ) of M&S
Bowl paper: (ξ,u,φ) = (ξ1,ξ2,ξ3) = (η,θ,ψ) of M&S
So at least I am reasonably consistent in the two papers.
Now in another of my docs ("toroidal coordinates.doc") where I am using these same (ξ,u,φ) coordinates, I write these equations:
x = a cosφ shξ/(chξ - cosu) ρ = a shξ/(chξ - cosu) ρ2 = x2+ y2
y = a sinφ shξ/(chξ - cosu)
z = a sinu/(chξ - cosu)
3. What toroidal coordinates do M&F use on their page 666? (Their only Volume 1 foray)
On this page, the coordinates have two different names:
set one: (ξ1,ξ2,ξ3) set two: (μ,θ,φ)
connection between the two MF coordinates: (from below)
ξ1 = ch(μ)
ξ2 = cos(θ)
ξ3 = cos(φ)
I can then translate their various equations to the (μ,θ,φ) coordinates:
x = a cosφ shμ /(chμ-cosθ)
y = a sinφ shμ /(chμ-cosθ)
z = a sinθ /(chμ-cosθ)
If I compare these to my own equations in Section 2 above, I find that
(μ,θ,φ)MF666 = (ξ,u,φ)PLbowl
Now let's take a look at the scale factors given below by MFp666:
h1 = - a [shμ]-1/(chμ-cosθ)
h2 = a [sinθ]-1/(chμ-cosθ)
h3 = a shμ [sinφ]-1/(chμ-cosθ)
These do not match mine because THESE hi are for the MF ξi coordinates, not for the MF (μ,θ,ε) coordinates. OK, I think I understand this now, and I don't really care about their ξi coordinates and thus I don't care about their Stackel matrix in those coordinates. I never comment on this anywhere, but perhaps I should.
4. What toroidal coordinates do MF use in Volume II?
Here is what is on page 1301: (and following pages deal with toroidals)
This is consistent with their Volume 1 stuff EXCEPT they have now renamed the second variable to be η instead of θ. So here they have (μ,η,φ) whereas in Volume 1 they had (μ,θ,φ). This is then the same as my coordinates (ξ,u,φ). The hi here agree with mine.
So OK, there is no other stuff in Volume 2, so this completes my survey.
(μ,η,φ)MFp1301 = (μ,θ,φ)MFp666 = (ξ,u,φ)PLbowl = (ξ1,ξ1,ξ3)PLstackel = (η,θ,ψ)MoonSp
(ξ1,ξ1,ξ3)MFp666 = totally different coordinates