Parabolic coordinates
Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas.
Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges.
Two-dimensional parabolic coordinates
Two-dimensional parabolic coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] are defined by the equations, in terms of Cartesian coordinates:
- [math]\displaystyle{ x = \sigma \tau }[/math]
- [math]\displaystyle{ y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right) }[/math]
The curves of constant [math]\displaystyle{ \sigma }[/math] form confocal parabolae
- [math]\displaystyle{ 2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2} }[/math]
that open upwards (i.e., towards [math]\displaystyle{ +y }[/math]), whereas the curves of constant [math]\displaystyle{ \tau }[/math] form confocal parabolae
- [math]\displaystyle{ 2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2} }[/math]
that open downwards (i.e., towards [math]\displaystyle{ -y }[/math]). The foci of all these parabolae are located at the origin.
The Cartesian coordinates [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] can be converted to parabolic coordinates by:
- [math]\displaystyle{ \sigma = \operatorname{sign}(x)\sqrt{\sqrt{x^{2} +y^{2}}-y} }[/math]
- [math]\displaystyle{ \tau = \sqrt{\sqrt{x^{2} +y^{2}}+y} }[/math]
Two-dimensional scale factors
The scale factors for the parabolic coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] are equal
- [math]\displaystyle{ h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}} }[/math]
Hence, the infinitesimal element of area is
- [math]\displaystyle{ dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau }[/math]
and the Laplacian equals
- [math]\displaystyle{ \nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + \frac{\partial^{2} \Phi}{\partial \tau^{2}} \right) }[/math]
Other differential operators such as [math]\displaystyle{ \nabla \cdot \mathbf{F} }[/math] and [math]\displaystyle{ \nabla \times \mathbf{F} }[/math] can be expressed in the coordinates [math]\displaystyle{ (\sigma, \tau) }[/math] by substituting the scale factors into the general formulae found in orthogonal coordinates.
Three-dimensional parabolic coordinates
The two-dimensional parabolic coordinates form the basis for two sets of three-dimensional orthogonal coordinates. The parabolic cylindrical coordinates are produced by projecting in the [math]\displaystyle{ z }[/math]-direction. Rotation about the symmetry axis of the parabolae produces a set of confocal paraboloids, the coordinate system of tridimensional parabolic coordinates. Expressed in terms of cartesian coordinates:
- [math]\displaystyle{ x = \sigma \tau \cos \varphi }[/math]
- [math]\displaystyle{ y = \sigma \tau \sin \varphi }[/math]
- [math]\displaystyle{ z = \frac{1}{2} \left(\tau^{2} - \sigma^{2} \right) }[/math]
where the parabolae are now aligned with the [math]\displaystyle{ z }[/math]-axis, about which the rotation was carried out. Hence, the azimuthal angle [math]\displaystyle{ \phi }[/math] is defined
- [math]\displaystyle{ \tan \varphi = \frac{y}{x} }[/math]
The surfaces of constant [math]\displaystyle{ \sigma }[/math] form confocal paraboloids
- [math]\displaystyle{ 2z = \frac{x^{2} + y^{2}}{\sigma^{2}} - \sigma^{2} }[/math]
that open upwards (i.e., towards [math]\displaystyle{ +z }[/math]) whereas the surfaces of constant [math]\displaystyle{ \tau }[/math] form confocal paraboloids
- [math]\displaystyle{ 2z = -\frac{x^{2} + y^{2}}{\tau^{2}} + \tau^{2} }[/math]
that open downwards (i.e., towards [math]\displaystyle{ -z }[/math]). The foci of all these paraboloids are located at the origin.
The Riemannian metric tensor associated with this coordinate system is
- [math]\displaystyle{ g_{ij} = \begin{bmatrix} \sigma^2+\tau^2 & 0 & 0\\0 & \sigma^2+\tau^2 & 0\\0 & 0 & \sigma^2\tau^2 \end{bmatrix} }[/math]
Three-dimensional scale factors
The three dimensional scale factors are:
- [math]\displaystyle{ h_{\sigma} = \sqrt{\sigma^2+\tau^2} }[/math]
- [math]\displaystyle{ h_{\tau} = \sqrt{\sigma^2+\tau^2} }[/math]
- [math]\displaystyle{ h_{\varphi} = \sigma\tau }[/math]
It is seen that the scale factors [math]\displaystyle{ h_{\sigma} }[/math] and [math]\displaystyle{ h_{\tau} }[/math] are the same as in the two-dimensional case. The infinitesimal volume element is then
- [math]\displaystyle{ dV = h_\sigma h_\tau h_\varphi\, d\sigma\,d\tau\,d\varphi = \sigma\tau \left( \sigma^{2} + \tau^{2} \right)\,d\sigma\,d\tau\,d\varphi }[/math]
and the Laplacian is given by
- [math]\displaystyle{ \nabla^2 \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left[ \frac{1}{\sigma} \frac{\partial}{\partial \sigma} \left( \sigma \frac{\partial \Phi}{\partial \sigma} \right) + \frac{1}{\tau} \frac{\partial}{\partial \tau} \left( \tau \frac{\partial \Phi}{\partial \tau} \right)\right] + \frac{1}{\sigma^2\tau^2}\frac{\partial^2 \Phi}{\partial \varphi^2} }[/math]
Other differential operators such as [math]\displaystyle{ \nabla \cdot \mathbf{F} }[/math] and [math]\displaystyle{ \nabla \times \mathbf{F} }[/math] can be expressed in the coordinates [math]\displaystyle{ (\sigma, \tau, \phi) }[/math] by substituting the scale factors into the general formulae found in orthogonal coordinates.
See also
- Parabolic cylindrical coordinates
- Orthogonal coordinate system
- Curvilinear coordinates
Bibliography
- Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. pp. 660. ISBN 0-07-043316-X.
- Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 185–186. https://archive.org/details/mathematicsofphy0002marg.
- Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. pp. 180. ASIN B0000CKZX7.
- Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 96.
- Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. pp. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting uk for ξk.
- Moon P, Spencer DE (1988). "Parabolic Coordinates (μ, ν, ψ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 34–36 (Table 1.08). ISBN 978-0-387-18430-2.
External links
- Hazewinkel, Michiel, ed. (2001), "Parabolic coordinates", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/p071170
- MathWorld description of parabolic coordinates
Original source: https://en.wikipedia.org/wiki/Parabolic coordinates.
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