Stäckel Determinant

A Determinant used to determine in which coordinate systems the Helmholtz Differential Equation is separable (Morse and Feshbach 1953). A determinant

$\begin{displaymath} S=\vert\Phi_{mn}\vert = \left\vert\matrix{ \Phi_{11} & \Phi_... ... \Phi_{23}\cr \Phi_{31} & \Phi_{32} & \Phi_{33}\cr}\right\vert \end{displaymath}$

(1)

in which $\Phi_{ni}$ are functions of

alone is called a Stäckel determinant. A coordinate system is separable if it obeys the Robertson Condition, namely that the Scale Factors

in the Laplacian

$\begin{displaymath} \nabla^2=\sum_{i=1}^3 {1\over h_1h_2h_3}{\partial\over\parti... ...ft({{h_1h_2h_3\over{h_i}^2}{\partial\over\partial u_i}}\right) \end{displaymath}$

(2)

can be rewritten in terms of functions

defined by

$\begin{displaymath} {1\over h_1h_2h_3}{\partial\over\partial u_i}\left({{h_1h_2h... ...over\partial u_i}\left({f_i{\partial\over\partial u_i}}\right) \end{displaymath}$

(3)

such that

can be written

$\begin{displaymath} S={h_1h_2h_3\over f_1(u_1)f_2(u_2)f_3(u_3)}. \end{displaymath}$

(4)

When this is true, the separated equations are of the form

$\begin{displaymath} {1\over f_n}{\partial\over\partial u_n}\left({f_n{\partial X... ...ght)+({k_1}^2\Phi_{n1}+{k_2}^2\Phi_{n2}+{k_3}^2\Phi_{n3})X_n=0 \end{displaymath}$

(5)

The $\Phi_{ij}$ s obey the minor equations

$\displaystyle M_1$	$\textstyle =$	$\displaystyle \Phi_{22}\Phi_{33}-\Phi_{23}\Phi_{32}={S\over h_1^2}$	(6)
$\displaystyle M_2$	$\textstyle =$	$\displaystyle \Phi_{13}\Phi_{31}-\Phi_{12}\Phi_{33}={S\over h_2^2}$	(7)
$\displaystyle M_3$	$\textstyle =$	$\displaystyle \Phi_{12}\Phi_{23}-\Phi_{13}\Phi_{22}={S\over h_3^2},$	(8)

which are equivalent to

$\begin{displaymath} M_1\Phi_{11}+M_2\Phi_{21}+M_3\Phi_{31}=S \end{displaymath}$

(9)

$\begin{displaymath} M_1\Phi_{12}+M_2\Phi_{22}+M_3\Phi_{32}=0 \end{displaymath}$

(10)

$\begin{displaymath} M_1\Phi_{13}+M_2\Phi_{23}+M_3\Phi_{33}=0. \end{displaymath}$

(11)

This gives a total of four equations in nine unknowns. Morse and Feshbach (1953, pp. 655-666) give not only the Stäckel determinants for common coordinate systems, but also the elements of the determinant (although it is not clear how these are derived).

References

Morse, P. M. and Feshbach, H. ``Tables of Separable Coordinates in Three Dimensions.'' Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 509-511 and 655-666, 1953.