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Date: 12-7-2018
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In spherical coordinates, the scale factors are , , , and the separation functions are , , , giving a Stäckel determinant of .
The Laplacian is
(1) |
To solve Laplace's equation in spherical coordinates, attempt separation of variables by writing
(2) |
Then the Helmholtz differential equation becomes
(3) |
Now divide by ,
(4) |
(5) |
The solution to the second part of (5) must be sinusoidal, so the differential equation is
(6) |
which has solutions which may be defined either as a complex function with , ...,
(7) |
or as a sum of real sine and cosine functions with , ...,
(8) |
Plugging (6) back into (7),
(9) |
The radial part must be equal to a constant
(10) |
(11) |
But this is the Euler differential equation, so we try a series solution of the form
(12) |
Then
(13) |
(14) |
(15) |
This must hold true for all powers of . For the term (with ),
(16) |
which is true only if and all other terms vanish. So for , . Therefore, the solution of the component is given by
(17) |
Plugging (17) back into (◇),
(18) |
(19) |
which is the associated Legendre differential equation for and , ..., . The general complex solution is therefore
(20) |
where
(21) |
are the (complex) spherical harmonics. The general real solution is
(22) |
Some of the normalization constants of can be absorbed by and , so this equation may appear in the form
(23) |
where
(24) |
(25) |
are the even and odd (real) spherical harmonics. If azimuthal symmetry is present, then is constant and the solution of the component is a Legendre polynomial . The general solution is then
(26) |
REFERENCES:
Byerly, W. E. An Elementary Treatise on Fourier's Series, and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics. New York: Dover, p. 244, 1959.
Moon, P. and Spencer, D. E. Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, p. 27, 1988.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 514 and 658, 1953.
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