On the surface of a Sphere, attempt Separation of Variables in Spherical Coordinates by writing
![\begin{displaymath}
F(\theta, \phi) = \Theta (\theta)\Phi(\phi),
\end{displaymath}](h_1157.gif) |
(1) |
then the Helmholtz Differential Equation becomes
![\begin{displaymath}
{1\over\sin^2\phi}{d^2\Theta\over d\theta^2}\Phi + {\cos\phi...
...hi}\Theta
+ {d^2\Phi\over d\phi^2}\Theta + k^2\Theta\Phi = 0.
\end{displaymath}](h_1158.gif) |
(2) |
Dividing both sides by
,
![\begin{displaymath}
\left({{\cos\phi\sin\phi\over\Phi} {d\Phi\over d\phi }+ {\si...
...ft({{1\over \Theta }{d^2\Theta\over d\theta^2}+k^2}\right)= 0,
\end{displaymath}](h_1160.gif) |
(3) |
which can now be separated by writing
![\begin{displaymath}
{d^2\Theta\over d\theta^2}{1\over\Theta}= -(k^2+m^2).
\end{displaymath}](h_1161.gif) |
(4) |
The solution to this equation must be periodic, so
must be an Integer. The solution may then be defined either as
a Complex function
![\begin{displaymath}
\Theta (\theta) = A_me^{i\sqrt{k^2+m^2}\,\theta}+B_me^{-i\sqrt{k^2+m^2}\,\theta}
\end{displaymath}](h_1162.gif) |
(5) |
for
, ...,
, or as a sum of Real sine and cosine functions
![\begin{displaymath}
\Theta(\theta) = S_m\sin(\sqrt{k^2+m^2}\,\theta)+C_m\cos(\sqrt{k^2+m^2}\,\theta)
\end{displaymath}](h_1163.gif) |
(6) |
for
, ...,
. Plugging (4) into (3) gives
![\begin{displaymath}
{\cos\phi\sin\phi\over\Phi}{d\Phi\over d\phi} + {\sin^2\phi\over\Phi}{d^2\Phi\over d\phi^2} + m^2 = 0
\end{displaymath}](h_1164.gif) |
(7) |
![\begin{displaymath}
\Phi'' + {\cos\phi\over\sin\phi}\Phi' + {m^2\over\sin^2\phi}\Phi = 0,
\end{displaymath}](h_1165.gif) |
(8) |
which is the Legendre Differential Equation for
with
![\begin{displaymath}
m^2\equiv l(l+1),
\end{displaymath}](h_1166.gif) |
(9) |
giving
![\begin{displaymath}
l^2+l-m^2=0
\end{displaymath}](h_1167.gif) |
(10) |
![\begin{displaymath}
l={\textstyle{1\over 2}}(-1\pm \sqrt{1+4m^2}\,).
\end{displaymath}](h_1168.gif) |
(11) |
Solutions are therefore Legendre Polynomials with a Complex index.
The general Complex solution is then
![\begin{displaymath}
F(\theta,\phi)=\sum_{m=-\infty}^\infty P_l(\cos \phi)(A_me^{im\theta}+B_me^{-im\theta}),
\end{displaymath}](h_1169.gif) |
(12) |
and the general Real solution is
![\begin{displaymath}
F(\theta,\phi)=\sum_{m=0}^\infty P_l(\cos\phi)[S_m\sin(m\theta)+C_m\cos(m\theta)].
\end{displaymath}](h_1170.gif) |
(13) |
Note that these solutions depend on only a single variable
. However, on the surface of a sphere, it is usual to express
solutions in terms of the Spherical Harmonics derived for the 3-D spherical case, which depend on
the two variables
and
.
© 1996-9 Eric W. Weisstein
1999-05-25