info prev up next book cdrom email home

Helmholtz Differential Equation--Confocal Ellipsoidal Coordinates

Using the Notation of Byerly (1959, pp. 252-253), Laplace's Equation can be reduced to

\end{displaymath} (1)

$\displaystyle \alpha$ $\textstyle =$ $\displaystyle c\int_c^\lambda {d\lambda\over\sqrt{(\lambda^2-b^2)(\lambda^2-c^2)}}$  
  $\textstyle =$ $\displaystyle F\left({{b\over c}, {\pi\over 2}}\right)-F\left({{b\over c}, \sin^{-1}\left({c\over\lambda}\right)}\right)$ (2)
$\displaystyle \beta$ $\textstyle =$ $\displaystyle c\int_b^\mu {d\mu\over\sqrt{(c^2-\mu^2)(\mu^2-b^2)}}$  
  $\textstyle =$ $\displaystyle F\left({\sqrt{1-{b^2-c^2}}, \sin^{-1}\left({\sqrt{1-{b^2\over\mu^2}\over 1-{b^2\over c^2}}\,}\right)}\right)$ (3)
$\displaystyle \gamma$ $\textstyle =$ $\displaystyle c\int_0^\nu{d\nu\over\sqrt{(b^2-\nu^2)(c^2-\nu^2)}}$  
  $\textstyle =$ $\displaystyle F\left({{b\over c}, \sin^{-1}\left({\nu\over b}\right)}\right).$ (4)

In terms of $\alpha$, $\beta$, and $\gamma$,
$\displaystyle \lambda$ $\textstyle =$ $\displaystyle c\mathop{\rm dc}\nolimits \left({\alpha, {b\over c}}\right)$ (5)
$\displaystyle \mu$ $\textstyle =$ $\displaystyle b\mathop{\rm nd}\nolimits \left({\beta,\sqrt{1-{b^2\over c^2}}\,}\right)$ (6)
$\displaystyle \nu$ $\textstyle =$ $\displaystyle b\mathop{\rm sn}\nolimits \left({\gamma,{b\over c}}\right).$ (7)

Equation (1) is not separable using a function of the form
\end{displaymath} (8)

but it is if we let
$\displaystyle {1\over L}{d^2L\over d\alpha^2}$ $\textstyle =$ $\displaystyle \sum a_k\lambda^k$ (9)
$\displaystyle {1\over M}{d^2M\over d\beta^2}$ $\textstyle =$ $\displaystyle \sum b_k\mu^k$ (10)
$\displaystyle {1\over N}{d^2N\over d\gamma^2}$ $\textstyle =$ $\displaystyle \sum c_k\nu^k.$ (11)

These give
$\displaystyle a_0$ $\textstyle =$ $\displaystyle -b_0=c_0$ (12)
$\displaystyle a_2$ $\textstyle =$ $\displaystyle -b_2=c_2,$ (13)

and all others terms vanish. Therefore (1) can be broken up into the equations
$\displaystyle {d^2L\over d\alpha^2}$ $\textstyle =$ $\displaystyle (a_0+a_2\lambda^2)L$ (14)
$\displaystyle {d^2M\over d\beta^2}$ $\textstyle =$ $\displaystyle -(a_0+a_2\mu^2)M$ (15)
$\displaystyle {d^2N\over d\gamma^2}$ $\textstyle =$ $\displaystyle (a_0+a_2\nu^2)N.$ (16)

For future convenience, now write
$\displaystyle a_0$ $\textstyle =$ $\displaystyle -(b^2+c^2)p$ (17)
$\displaystyle a_2$ $\textstyle =$ $\displaystyle m(m+1),$ (18)

$\displaystyle {d^2L\over d\alpha^2}-[m(m+1)\lambda^2-(b^2+c^2)p]L$ $\textstyle =$ $\displaystyle 0$ (19)
$\displaystyle {d^2M\over d\beta^2}+[m(m+1)\mu^2-(b^2+c^2)p]M$ $\textstyle =$ $\displaystyle 0$ (20)
$\displaystyle {d^2N\over d\gamma^2}-[m(m+1)\nu^2-(b^2+c^2)p]N$ $\textstyle =$ $\displaystyle 0.$ (21)

Now replace $\alpha$, $\beta$, and $\gamma$ to obtain

$\displaystyle (\lambda^2-b^2)(\lambda^2-c^2){d^2L\over d\lambda^2}+\lambda(\lambda^2-b^2+\lambda^2-c^2){dL\over d\lambda}-[m(m+1)\lambda^2-(b^2+c^2)p]L$ $\textstyle =$ $\displaystyle 0$ (22)
$\displaystyle (\mu^2-b^2)(\mu^2-c^2){d^2M\over d\mu^2}+\mu(\mu^2-b^2+\mu^2-c^2){dM\over d\mu}-[m(m+1)\mu^2-(b^2+c^2)p]M$ $\textstyle =$ $\displaystyle 0$ (23)
$\displaystyle (\nu^2-b^2)(\nu^2-c^2){d^2N\over d\nu^2}+\nu(\nu^2-b^2+\nu^2-c^2){dN\over d\nu}-[m(m+1)\nu^2-(b^2+c^2)p]N$ $\textstyle =$ $\displaystyle 0.$ (24)

Each of these is a Lamé's Differential Equation, whose solution is called an Ellipsoidal Harmonic. Writing
$\displaystyle L(\lambda)$ $\textstyle =$ $\displaystyle E_m^p(\lambda)$ (25)
$\displaystyle M(\lambda)$ $\textstyle =$ $\displaystyle E_m^p(\mu)$ (26)
$\displaystyle N(\lambda)$ $\textstyle =$ $\displaystyle E_m^p(\nu)$ (27)

gives the solution to (1) as a product of Ellipsoidal Harmonics $E_m^p(x)$.
\end{displaymath} (28)


Arfken, G. ``Confocal Ellipsoidal Coordinates $(\xi_1, \xi_2, \xi_3)$.'' §2.15 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 117-118, 1970.

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, pp. 251-258, 1959.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 663, 1953.

info prev up next book cdrom email home

© 1996-9 Eric W. Weisstein