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

$$x = {\lambda\mu\nu\over ab}$$ (1)

$$y = {\lambda\over a} \sqrt{(\mu^2-a^2)(\nu^2-a^2)\over a^2-b^2}$$ (2)

$$z = {\lambda\over b} \sqrt{(\mu^2-b^2)(\nu^2-b^2)\over b^2-a^2},$$ (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

$$x^2+y^2+z^2=\lambda^2$$ (4)

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

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

The Scale Factors are

$${h_\lambda } = 1$$ (7)

$${h_\mu} = \sqrt{\lambda^2(\mu^2-\nu^2)\over(\mu^2-a^2)(b^2-\mu^2)}$$ (8)

$${h_\nu} = \sqrt{\lambda^2(\mu^2-\nu^2)\over(\nu^2-a^2)(\nu^2-b^2)}.$$ (9)

The Laplacian is

$\nabla^2 = {\nu(2\nu^2-a^2-b^2)\over(\mu-\nu)(\mu+\nu)\lambda^2}{\partial\over\... ...u)(\nu-b)(\nu+b)\over(\nu-\mu)(\nu+\mu)\lambda^2}{\partial^2\over\partial\nu^2}$
$+{\mu(2\mu^2-a^2-b^2)\over(\nu-\mu)(\nu+\mu)\lambda^2}{\partial\over\partial\m... ...\lambda}{\partial\over\partial\lambda}+{\partial^2\over\partial\lambda^2}.\quad$	(10)

The Helmholtz Differential Equation is separable in conical coordinates.

See also Helmholtz Differential Equation--Conical Coordinates

References

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.