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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

del ^2=sum_(i=1)^31/(h_1h_2h_3)partial/(partialu_i)((h_1h_2h_3)/(h_i^2)partial/(partialu_i))
(1)

and the functions f_i(u_i) and Phi_(ij) defined by

1/(f_n)partial/(partialu_n)(f_n(partialX_n)/(partialu_n))+(k_1^2Phi_(n1)+k_2^2Phi_(n2)+k_3^2Phi_(n3))X_n=0
(2)

must be of the form of a Stäckel determinant

S=|Phi_(mn)|=|Phi_(11) Phi_(12) Phi_(13); Phi_(21) Phi_(22) Phi_(23); Phi_(31) Phi_(32) Phi_(33)|=(h_1h_2h_3)/(f_1(u_1)f_2(u_2)f_3(u_3)).
(3)

See also

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

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References

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

Referenced on Wolfram|Alpha

Robertson Condition

Cite this as:

Weisstein, Eric W. "Robertson Condition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RobertsonCondition.html

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