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Parabolic coordinates

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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.

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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 ( σ , τ ) are defined by the equations, in terms of cartesian coordinates:

x = σ τ y = 1 2 ( τ 2 − σ 2 )

The curves of constant σ form confocal parabolae

2 y = x 2 σ 2 − σ 2

that open upwards (i.e., towards + y ), whereas the curves of constant τ form confocal parabolae

2 y = − x 2 τ 2 + τ 2

that open downwards (i.e., towards − y ). The foci of all these parabolae are located at the origin.

Two-dimensional scale factors

The scale factors for the parabolic coordinates ( σ , τ ) are equal

h σ = h τ = σ 2 + τ 2

Hence, the infinitesimal element of area is

d A = ( σ 2 + τ 2 ) d σ d τ

and the Laplacian equals

∇ 2 Φ = 1 σ 2 + τ 2 ( ∂ 2 Φ ∂ σ 2 + ∂ 2 Φ ∂ τ 2 )

Other differential operators such as ∇ ⋅ F and ∇ × F can be expressed in the coordinates ( σ , τ ) 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 z -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:

x = σ τ cos ⁡ φ y = σ τ sin ⁡ φ z = 1 2 ( τ 2 − σ 2 )

where the parabolae are now aligned with the z -axis, about which the rotation was carried out. Hence, the azimuthal angle ϕ is defined

tan ⁡ φ = y x

The surfaces of constant σ form confocal paraboloids

2 z = x 2 + y 2 σ 2 − σ 2

that open upwards (i.e., towards + z ) whereas the surfaces of constant τ form confocal paraboloids

2 z = − x 2 + y 2 τ 2 + τ 2

that open downwards (i.e., towards − z ). The foci of all these paraboloids are located at the origin.

The Riemannian metric tensor associated with this coordinate system is

g i j = [ σ 2 + τ 2 0 0 0 σ 2 + τ 2 0 0 0 σ 2 τ 2 ]

Three-dimensional scale factors

The three dimensional scale factors are:

h σ = σ 2 + τ 2 h τ = σ 2 + τ 2 h φ = σ τ

It is seen that The scale factors h σ and h τ are the same as in the two-dimensional case. The infinitesimal volume element is then

d V = h σ h τ h φ d σ d τ d φ = σ τ ( σ 2 + τ 2 ) d σ d τ d φ

and the Laplacian is given by

∇ 2 Φ = 1 σ 2 + τ 2 [ 1 σ ∂ ∂ σ ( σ ∂ Φ ∂ σ ) + 1 τ ∂ ∂ τ ( τ ∂ Φ ∂ τ ) ] + 1 σ 2 τ 2 ∂ 2 Φ ∂ φ 2

Other differential operators such as ∇ ⋅ F and ∇ × F can be expressed in the coordinates ( σ , τ , ϕ ) by substituting the scale factors into the general formulae found in orthogonal coordinates.

References

Parabolic coordinates Wikipedia
(Text) CC BY-SA


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