Toroidal Coordinates

$\begin{figure}\begin{center}\BoxedEPSF{ToroidalCoordinates.epsf scaled 1000}\end{center}\end{figure}$

A system of Curvilinear Coordinates for which several different notations are commonly used. In this work (, , $\phi$ ) is used, whereas Arfken (1970) uses ( $\xi$ , $\eta$ , $\varphi$ ). The toroidal coordinates are defined by

$\displaystyle x$	$\textstyle =$	$\displaystyle {a\sinh u\cos\phi\over\cosh u-\cos v}$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle {a\sinh u\sin\phi\over\cosh u-\cos v}$	(2)
$\displaystyle z$	$\textstyle =$	$\displaystyle {a\sin v\over \cosh u-\cos v},$	(3)

where $\sinh z$ is the Hyperbolic Sine and $\cosh z$ is the Hyperbolic Cosine. The Scale Factors are

$\displaystyle h_u$	$\textstyle =$	$\displaystyle {a\over\cosh u-\cos v}$	(4)
$\displaystyle h_v$	$\textstyle =$	$\displaystyle {a\over\cosh u-\cos v}$	(5)
$\displaystyle h_\phi$	$\textstyle =$	$\displaystyle {a\sinh u\over\cosh u-\cos v}.$	(6)

The Laplacian is

$\nabla^2f={\sinh u\over(\cosh u-\cos v)^3}\left[{{\partial\over\partial u}\left({{\sinh u\over\cosh u-\cos v}{\partial f\over\partial u}}\right)}\right.$

$+\left.{{\partial\over\partial v}\left({{\sinh u\over\cosh u-\cos v}{\partial ... ...olimits u\over\cosh u-\cos v}{\partial f\over\partial\phi}}\right)}\right]\quad$ (7)

$= (\cos v-\cosh u)\left[{\sin v{\partial f\over\partial v}+(\cos v-\cosh u)\lef... ...partial^2 f\over\partial\phi^2}+{\partial^2 f\over\partial v^2}}\right)}\right.$

$+\left.{(\cos v\cosh u-1)\mathop{\rm csch}\nolimits u{\partial f\over\partial u}+(\cos v-\cosh u){\partial^2f\over\partial u^2}}\right].\quad$ (8)

The Helmholtz Differential Equation is not separable in toroidal coordinates, but Laplace's Equation is.

References

Arfken, G. ``Toroidal Coordinates ( $\xi$ , $\eta$ , $\phi$ ).'' §2.13 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 112-115, 1970.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 666, 1953.

$\nabla^2f={\sinh u\over(\cosh u-\cos v)^3}\left[{{\partial\over\partial u}\left({{\sinh u\over\cosh u-\cos v}{\partial f\over\partial u}}\right)}\right.$
$+\left.{{\partial\over\partial v}\left({{\sinh u\over\cosh u-\cos v}{\partial ... ...olimits u\over\cosh u-\cos v}{\partial f\over\partial\phi}}\right)}\right]\quad$	(7)
$= (\cos v-\cosh u)\left[{\sin v{\partial f\over\partial v}+(\cos v-\cosh u)\lef... ...partial^2 f\over\partial\phi^2}+{\partial^2 f\over\partial v^2}}\right)}\right.$
$+\left.{(\cos v\cosh u-1)\mathop{\rm csch}\nolimits u{\partial f\over\partial u}+(\cos v-\cosh u){\partial^2f\over\partial u^2}}\right].\quad$	(8)