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 divide by ,
|
(4) |
Separating the part,
|
(5) |
|
(6) |
so
|
(7) |
which has solution
|
(8) |
and
|
(9) |
|
(10) |
This can be separated
|
(11) |
|
(12) |
so
|
(13) |
|
(14) |
References
Arfken, G. ``Parabolic Coordinates
.'' §2.12 in
Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 109-111, 1970.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York:
McGraw-Hill, pp. 514-515 and 660, 1953.
© 1996-9 Eric W. Weisstein
1999-05-25