Parabolic coordinates

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 (sigma, tau) are defined by the equations

x = sigma tau,

y = frac{1}{2} left(tau^{2} - sigma^{2} right)

The curves of constant sigma form confocal parabolae

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

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

2y = -frac{x^{2}}{tau^{2}} + tau^{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 (sigma, tau) are equal

h_{sigma} = h_{tau} = sqrt{sigma^{2} + tau^{2}}

Hence, the infinitesimal element of area is

dA = left(sigma^{2} + tau^{2} right) dsigma dtau

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)

Other differential operators such as nabla cdot mathbf{F} and nabla times mathbf{F} can be expressed in the coordinates (sigma, tau) 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, forming a coordinate system that is also known as "parabolic coordinates"

x = sigma tau cos phi

y = sigma tau sin phi

z = frac{1}{2} left(tau^{2} - sigma^{2} right)

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

tan phi = frac{y}{x}

The surfaces of constant sigma form confocal paraboloids

2z = frac{x^{2} + y^{2}}{sigma^{2}} - sigma^{2}

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

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

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

Three-dimensional scale factors

The three dimensional scale factors are:

h_{sigma} = sqrt{sigma^2+tau^2}
h_{tau} = sqrt{sigma^2+tau^2}
h_{phi} = sigmatau,

It is seen that The scale factors h_{sigma} and h_{tau} are the same as in the two-dimensional case. The infinitesimal volume element is then

dV = h_sigma h_tau h_phi = sigmatau left(sigma^{2} + tau^{2} right),dsigma,dtau,dphi

and the Laplacian is given by

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^2tau^2}frac{partial^2 Phi}{partial phi^2}

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

An alternative formulation

Conversion from Cartesian to parabolic coordinates is affected by means of the following equations:

xi = sqrt{sqrt{ x^2 + y^2 + z^2 } + z},
eta = sqrt{sqrt{ x^2 + y^2 + z^2 } - z},
phi = arctan {y over x}.

begin{vmatrix}detadxidphiend{vmatrix} = begin{vmatrix} frac{x}{sqrt{x^2+y^2+z^2}} & frac{y}{sqrt{x^2+y^2+z^2}} &-1+frac{z}{sqrt{x^2+y^2+z^2}} frac{x}{sqrt{x^2+y^2+z^2}} & frac{y}{sqrt{x^2+y^2+z^2}} &1 +frac{z}{sqrt{x^2+y^2+z^2}} frac{-y}{x^2+y^2}&frac{x}{x^2+y^2}&0 end{vmatrix} cdot begin{vmatrix}dxdydzend{vmatrix}

etage 0,quadxige 0

If φ=0 then a cross-section is obtained; the coordinates become confined to the x-z plane:

eta = -z + sqrt{ x^2 + z^2},
xi = z + sqrt{ x^2 + z^2}.

If η=c (a constant), then

left. z right|_{eta = c} = {x^2 over 2 c} - {c over 2}.
This is a parabola whose focus is at the origin for any value of c. The parabola's axis of symmetry is vertical and the concavity faces upwards.

If ξ=c then

left. z right|_{xi = c} = {c over 2} - {x^2 over 2 c}.
This is a parabola whose focus is at the origin for any value of c. Its axis of symmetry is vertical and the concavity faces downwards.

Now consider any upward parabola η=c and any downward parabola ξ=b. It is desired to find their intersection:

{x^2 over 2 c} - {c over 2} = {b over 2} - {x^2 over 2 b},
regroup,
{x^2 over 2 c} + {x^2 over 2 b} = {b over 2} + {c over 2},
factor out the x,
x^2 left({b + c over 2 b c} right) = {b + c over 2},
cancel out common factors from both sides,
x^2 = b c, ,
take the square root,
x = sqrt{b c}.
x is the geometric mean of b and c. The abscissa of the intersection has been found. Find the ordinate. Plug in the value of x into the equation of the upward parabola:
z_c = {b c over 2 c} - {c over 2} = {b - c over 2},
then plug in the value of x into the equation of the downward parabola:
z_b = {b over 2} - {b c over 2 b} = {b - c over 2}.
zc = zb, as should be. Therefore the point of intersection is
P : left(sqrt{b c}, {b - c over 2} right).

Draw a pair of tangents through point P, each one tangent to each parabola. The tangential line through point P to the upward parabola has slope:

{d z_c over d x} = {x over c} = { sqrt{ b c} over c} = sqrt{ b over c} = s_c.
The tangent through point P to the downward parabola has slope:
{d z_b over d x} = - {x over b} = { - sqrt{ b c } over b} = - sqrt{ {c over b} } = s_b.

The products of the two slopes is

s_c s_b = - sqrt{ {b over c}} sqrt{ {c over b}} = -1.
The product of the slopes is negative one, therefore the slopes are perpendicular. This is true for any pair of parabolas with concavities in opposite directions.

Such a pair of parabolas intersect at two points, but when φ is restricted to zero, it actually confines the other coordinates η and ξ to move in a half-plane with x>0, because x<0 corresponds to φ=π.

Thus a pair of coordinates η and ξ specify a unique point on the half-plane. Then letting φ range from 0 to 2π the half-plane revolves with the point (around the z-axis as its hinge): the parabolas form paraboloids. A pair of opposing paraboloids specifies a circle, and a value of φ specifies a half-plane which cuts the circle of intersection at a unique point. The point's Cartesian coordinates are [Menzel, p. 139]:

x = sqrt{xi eta} cos phi,
y = sqrt{xi eta} sin phi,
z = begin{matrix}frac{1}{2}end{matrix} (xi - eta ).

begin{vmatrix}dxdydzend{vmatrix} = begin{vmatrix} frac{1}{2}sqrt{frac{xi}{eta}}cosphi &frac{1}{2}sqrt{frac{eta}{xi}}cosphi &-sqrt{xieta}sinphi frac{1}{2}sqrt{frac{xi}{eta}}sinphi &frac{1}{2}sqrt{frac{eta}{xi}}sinphi &sqrt{xieta}cosphi -frac{1}{2}&frac{1}{2}&0 end{vmatrix} cdot begin{vmatrix}detadxidphiend{vmatrix}

See also

Bibliography

  • Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 0-07-043316-X,
  • Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand.
  • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill.
  • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag.
  • Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. Same as Morse & Feshbach (1953), substituting uk for ξk.
  • Moon P, Spencer DE (1988). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions. corrected 2nd ed., 3rd print ed., New York: Springer-Verlag.

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