释义 |
Helmholtz Differential Equation--Parabolic CoordinatesThe 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. |