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Conical Coordinates

Arfken (1970) and Morse and Feshbach (1953) use slightly different definitions of these coordinates. The system used in Mathematica ${}^{\scriptstyle\circledRsymbol}$ (Wolfram Research, Inc., Champaign, Illinois) is

$\displaystyle x$ $\textstyle =$ $\displaystyle {\lambda\mu\nu\over ab}$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle {\lambda\over a} \sqrt{(\mu^2-a^2)(\nu^2-a^2)\over a^2-b^2}$ (2)
$\displaystyle z$ $\textstyle =$ $\displaystyle {\lambda\over b} \sqrt{(\mu^2-b^2)(\nu^2-b^2)\over b^2-a^2},$ (3)

where $b^2>\mu^2>c^2>\nu^2$. The Notation of Byerly replaces $\lambda$ with $r$, and $a$ and $b$ with $b$ and $c$. The above equations give
\end{displaymath} (4)

{x^2\over\mu^2}+{y^2\over \mu^2-a^2}+{z^2\over\mu^2-b^2}=0
\end{displaymath} (5)

{x^2\over\nu^2}+{y^2\over \nu^2-a^2}+{z^2\over\nu^2-b^2}=0.
\end{displaymath} (6)

The Scale Factors are
$\displaystyle {h_\lambda }$ $\textstyle =$ $\displaystyle 1$ (7)
$\displaystyle {h_\mu}$ $\textstyle =$ $\displaystyle \sqrt{\lambda^2(\mu^2-\nu^2)\over(\mu^2-a^2)(b^2-\mu^2)}$ (8)
$\displaystyle {h_\nu}$ $\textstyle =$ $\displaystyle \sqrt{\lambda^2(\mu^2-\nu^2)\over(\nu^2-a^2)(\nu^2-b^2)}.$ (9)

The Laplacian is

$\nabla^2 = {\nu(2\nu^2-a^2-b^2)\over(\mu-\nu)(\mu+\nu)\lambda^2}{\partial\over\...
$ +{\mu(2\mu^2-a^2-b^2)\over(\nu-\mu)(\nu+\mu)\lambda^2}{\partial\over\partial\m...
...\lambda}{\partial\over\partial\lambda}+{\partial^2\over\partial\lambda^2}.\quad$ (10)
The Helmholtz Differential Equation is separable in conical coordinates.

See also Helmholtz Differential Equation--Conical Coordinates


Arfken, G. ``Conical Coordinates ($\xi_1$, $\xi_2$, $\xi_3$).'' §2.16 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 118-119, 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, p. 263, 1959.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 659, 1953.

Spence, R. D. ``Angular Momentum in Sphero-Conal Coordinates.'' Amer. J. Phys. 27, 329-335, 1959.

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