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Robertson Condition

For the Helmholtz Differential Equation to be Separable in a coordinate system, the Scale Factors $h_i$ in the Laplacian

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

and the functions $f_i(u_i)$ and $\Phi_{ij}$ defined by
{1\over f_n}{\partial\over\partial u_n}\left({f_n{\partial X...
\end{displaymath} (2)

must be of the form of a Stäckel Determinant
S=\vert\Phi_{mn}\vert = \left\vert\matrix{
\Phi_{11} & \Phi_...
...{33}\cr}\right\vert={h_1h_2h_3\over f_1(u_1)f_2(u_2)f_3(u_3)}.
\end{displaymath} (3)

See also Helmholtz Differential Equation, Laplace's Equation, Separation of Variables, Stäckel Determinant


Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part 1. New York: McGraw-Hill, p. 510, 1953.

© 1996-9 Eric W. Weisstein