The scale factors are , and the separation functions are , , , given a Stäckel
determinant of . The Laplacian is
|
(1)
|
Attempt separation of variables by writing
|
(2)
|
then the Helmholtz differential equation
becomes
|
(3)
|
Now multiply through by ,
|
(4)
|
Separating the part gives
|
(5)
|
which has solution
|
(6)
|
Plugging (5) back into (4) and multiplying by gives
|
(7)
|
Rewriting,
|
(8)
|
This can be rearranged into two terms, each containing only or ,
|
(9)
|
and so can be separated by letting the first part equal and the second equal , giving
|
(10)
|
|
(11)
|
See also
Helmholtz Differential
Equation,
Parabolic Coordinates
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References
Arfken, G. "Parabolic Coordinates ." §2.12 in Mathematical
Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-111,
1970.Moon, P. and Spencer, D. E. Field
Theory Handbook, Including Coordinate Systems, Differential Equations, and Their
Solutions, 2nd ed. New York: Springer-Verlag, p. 36, 1988.Morse,
P. M. and Feshbach, H. Methods
of Theoretical Physics, Part I. New York McGraw-Hill, pp. 514-515 and
660, 1953.
Cite this as:
Weisstein, Eric W. "Helmholtz Differential Equation--Parabolic Coordinates." From MathWorld--A Wolfram
Web Resource. https://mathworld.wolfram.com/HelmholtzDifferentialEquationParabolicCoordinates.html
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