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Helmholtz Differential Equation--Parabolic Coordinates

The Scale Factors are $h_u=h_v=\sqrt{u^2+v^2}$, $h_\theta=uv$ and the separation functions are $f_1(u)=u$, $f_2(v)=v$, $f_3(\theta)=1$, given a Stäckel Determinant of $S=u^2+v^2$. The Laplacian is


\begin{displaymath}
{1\over u^2+v^2}\left({{1\over u}{\partial F\over \partial u...
...t)+{1\over u^2v^2}{\partial^2 F\over \partial \theta^2}+k^2=0.
\end{displaymath} (1)

Attempt Separation of Variables by writing
\begin{displaymath}
F(u,v,z)\equiv U(u)V(v)\Theta(\theta),
\end{displaymath} (2)

then the Helmholtz Differential Equation becomes


\begin{displaymath}
{1\over u^2+v^2}\left[{V\Theta \left({{1\over u}{dU\over du}...
...{dV\over dv}+{d^2V\over dv^2}}\right)}\right]+k^2 UV\Theta =0.
\end{displaymath} (3)

Now divide by $UV\Theta$,


\begin{displaymath}
{u^2v^2\over u^2+v^2}\left[{{1\over U}\left({{1\over u}{dU\o...
...ght)}\right]+{1\over \Theta} {d^2\Theta\over d\theta^2}+k^2=0.
\end{displaymath} (4)

Separating the $\Theta$ part,
\begin{displaymath}
{1\over \Theta}{d^2 \Theta\over f\theta^2}=-(k^2+m^2)
\end{displaymath} (5)


\begin{displaymath}
{u^2v^2\over u^2+v^2}\left[{{1\over U}\left({{1\over u}{dU\o...
...{{1\over v}{dV\over dv}+{d^2V\over dv^2}}\right)}\right]= k^2,
\end{displaymath} (6)

so
\begin{displaymath}
{d^2\Theta\over d\theta^2}=-(k^2+m^2)\Theta,
\end{displaymath} (7)

which has solution
\begin{displaymath}
\Theta(\theta )= A\cos(\sqrt{k^2+m^2}\,\theta)+B\sin(\sqrt{k^2+m^2}\,\theta),
\end{displaymath} (8)

and


\begin{displaymath}
\left[{{1\over U}\left({{1\over u}{dU\over du}+{d^2 U\over d...
...}+{d^2 V\over dv^2}}\right)}\right]-k^2{u^2+v^2\over u^2v^2}=0
\end{displaymath} (9)


\begin{displaymath}
\left[{{1\over U}\left({{1\over u}{dU\over du}+{d^2 U\over d...
...V\over dv}+{d^2 V\over dv^2}}\right)-{k^2\over v^2}}\right]=0.
\end{displaymath} (10)

This can be separated
\begin{displaymath}
{1\over U}\left({{1\over u}{dU\over du}+{d^2 U\over du^2}}\right)-{k^2\over u^2}=c
\end{displaymath} (11)


\begin{displaymath}
{1\over V}\left({{1\over v}{dV\over dv}+{d^2 V\over dv^2}}\right)-{k^2\over v^2}=-c,
\end{displaymath} (12)

so
\begin{displaymath}
u^2{d^2U\over du^2}+{dU\over du}-(c+k^2)U=0
\end{displaymath} (13)


\begin{displaymath}
v^2{d^2V\over dv^2}+{dV\over dv}+(c-k^2)V=0.
\end{displaymath} (14)


References

Arfken, G. ``Parabolic Coordinates $(\xi, \eta, \phi)$.'' §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.



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© 1996-9 Eric W. Weisstein
1999-05-25