268
23 Coordinate Systems
The kinetic energy ˆ
T in Jacobi coordinates reads
ˆ
T = −
¯
h
2
2Nm
Δ R −
N−1
k=1
¯
h
2
2μ k
Δ x k , μ k =
k
k + 1
m ,
(23.32)
with m the particle mass.
Hyperspherical Coordinates Hyperspherical coordinates generalize polar and
spherical coordinates to N dimensions via
x 1 = r cos θ 1
0 ≤ θ 1 · · · θ N−2 ≤ π
x 2 = r sin θ 1 cos θ 2
x 3 = r sin θ 1 sin θ 2 cos θ 3
. . .
x N = r sin θ 1 sin θ 2 · · · sin θ N−1
− π ≤ θ N−1 ≤ π
,
(23.33)
thus hyperspherical coordinates hold
N
i=1
x
2
i = r
2 .
(23.34)
23.2 Programs and Computational Aspects
23.2.1 The SPECFUNPHYS Class CoordTrafo
The transformation of a coordinate system to cartesian coordinates is based on
the equations listed above. The inverse transformation or the transformation to a
different coordinate system than the Cartesian is usually straightforward. Thus I
will only mention a few cases.
Elliptic Cylindrical Coordinates The transformation to cartesian coordinates is
given by Eqs. (23.19). The inverse transformation can be computed via
˜
x + i ˜
y = cosh α cos β + i · sinh α sin β with d · ( ˜
x, ˜
y) = (x, y) (23.35a)
= cosh(α + iβ)
thus
α + iβ = acosh( ˜
x + i ˜
y).
(23.35b)
Paraboloidal Coordinates The transformation to cartesian coordinates is given
by Eqs. (23.23). No simple inverse transformation was found, thus the inverse
transformation is based on a numerical ansatz using the MATLAB function fzero.