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}](s3_474.gif) |
(1) |
in which
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}](s3_478.gif) |
(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}](s3_480.gif) |
(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}](s3_481.gif) |
(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}](s3_482.gif) |
(5) |
The
s obey the minor equations
which are equivalent to
![\begin{displaymath}
M_1\Phi_{11}+M_2\Phi_{21}+M_3\Phi_{31}=S
\end{displaymath}](s3_490.gif) |
(9) |
![\begin{displaymath}
M_1\Phi_{12}+M_2\Phi_{22}+M_3\Phi_{32}=0
\end{displaymath}](s3_491.gif) |
(10) |
![\begin{displaymath}
M_1\Phi_{13}+M_2\Phi_{23}+M_3\Phi_{33}=0.
\end{displaymath}](s3_492.gif) |
(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).
See also Helmholtz Differential Equation, Laplace's Equation, Poisson's Equation,
Robertson Condition, Separation of Variables
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.
© 1996-9 Eric W. Weisstein
1999-05-26