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Parabolic coordinate system showing curves of constant σ and τ the horizontal and vertical axes are the x and y coordinates respectively. These coordinates are projected along the z-axis, and so this diagram will hold for any value of the z coordinate.
Parabolic coordinate system showing curves of constant σ and τ the horizontal and vertical axes are the x and y coordinates respectively. These coordinates are projected along the z-axis, and so this diagram will hold for any value of the z coordinate.

Parabolic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinate system in the perpendicular z-direction. Hence, the coordinate surfaces are confocal parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges.

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[edit] Basic definition

The parabolic cylindrical coordinates (σ,τ,z) are defined in terms of the Cartesian coordinates (x,y,z)  by:

x = \sigma \tau\,
y = \textstyle\frac{1}{2} \left( \tau^{2} - \sigma^{2} \right)
z = z\,

The surfaces of constant σ form confocal parabolic cylinders

2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2}

that open towards + y, whereas the surfaces of constant τ form confocal parabolic cylinders

2y = -\frac{x^{2}}{\tau^{2}} + \tau^{2}

that open in the opposite direction, i.e., towards − y. The foci of all these parabolic cylinders are located along the line defined by x = y = 0. The radius r has a simple formula as well

r = \sqrt{x^{2} + y^{2}} = \textstyle\frac{1}{2} \left( \sigma^{2} + \tau^{2} \right)

that proves useful in solving the Hamilton-Jacobi equation in parabolic coordinates for the inverse-square central force problem of mechanics; for further details, see the Laplace-Runge-Lenz vector article.

[edit] Scale factors

The scale factors for the parabolic cylindrical coordinates σ and τ are:

h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}
h_{z}=1\,

The infinitesimal element of volume is

dV = h_\sigma h_\tau h_z=\left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau dz

and the Laplacian equals

\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) + \frac{\partial^{2} \Phi}{\partial z^{2}}

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (σ,τ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

[edit] Parabolic Cylindrical harmonics

Since all of the surfaces of constant σ, τ and z  are conicoid, Laplace's equation is separable in parabolic cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written:

\Phi=S(\sigma)\,T(\tau)\,Z(z)

and Laplace's equation, divided by Φ, is written:

\frac{1}{\sigma^2+\tau^2} \left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right]+\frac{\ddot{Z}}{Z}=0

Since the Z  equation is separate from the rest, we may write

\frac{\ddot{Z}}{Z}=m^2

where m  in general, a complex constant. Z(z) has the solution:

Z_m(z)=A_{1m}\,e^{mz}+A_{2m}\,e^{-mz}\,

Where A1m and A2m are constants. Substituting − m2 for \ddot{Z}/Z , Laplace's equation may now be written:

\left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right]=m^2(\sigma^2+\tau^2)

We may now separate the S  and T  functions and introduce a function of m to obtain:

\frac{\ddot{S}}{S}-m^2\sigma^2=-2m(n+\textstyle\frac{1}{2})
\frac{\ddot{T}}{T}-m^2\tau^2 =+2m(n+\textstyle\frac{1}{2})


\ddot{S} - (m^2\sigma^2-2m(n+\textstyle\frac{1}{2}))S=0
\ddot{T} - (m^2\tau^2 +2m(n+\textstyle\frac{1}{2}))T=0


use w=\sigma\sqrt{2m} and an = − n − 1 / 2

\ddot{S} - (\textstyle\frac{1}{4}w^2+a)S=0

use w=\tau e^{i\pi/4}\sqrt{2m} and an = − n − 1 / 2

\ddot{T} + (\textstyle\frac{1}{4}w^2 -ia)T=0




The solutions to these equations are the two distinct types parabolic cylinder functions

S_{mn}(\sigma) = A_3\,U(a_n,\sigma\sqrt{2m}) + A_4\,V(a_n,\sigma\sqrt{2m})
T_{mn}(\tau) = A_5\,U(-a_n,\tau \sqrt{2m}) + A_6\,V(-a_n,\tau \sqrt{2m})

The parabolic cylinder harmonics for (m,n) are now the product of the solutions. The combination will reduce the number of constants to four and the general solution to Laplace's equation may be written:

\Phi(\sigma,\tau,z)=\sum_{m,n} A_{mn} S_{mn} T_{mn} Z_m\,

It is very useful to note that

U(a_n,z)=D_n(z)=(n!\sqrt{\pi})^{1/2}\psi_n(z/\sqrt{2})

where Dn(z) is the Whittaker function and ψn(z) is the Hermite function since the orthogonality relationship may be introduced:

\int_{-\infty}^\infty\psi_m(z)\psi_n(z)dz=\delta_{mn}

for n and m non-negative integers. Solutions to Laplace's equation which vanish at infinity may be written:

\Phi(\sigma,\tau,z)= \sum_{m=0}^\infty\sum_{n=0}^\infty (A_{mn}\cos m\phi + B_{mn}\sin m\phi) D_n(\sigma\sqrt{2m})\,D_n(\tau e^{i\pi/4}\sqrt{2m})\,

T is not real !!!!!!!

[edit] Applications

The classic applications of parabolic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which such coordinates allow a separation of variables. A typical example would be the electric field surrounding a flat semi-infinite conducting plate.

[edit] See also

[edit] References

  • Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
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