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Elliptic Cylindrical Coordinates

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EllipticCylindricalCoord

The v coordinates are the asymptotic angle of confocal hyperbolic cylinders symmetrical about the x-axis. The u coordinates are confocal elliptic cylinders centered on the origin.

x=acoshucosv
(1)
y=asinhusinv
(2)
z=z,
(3)

where u in [0,infty), v in [0,2pi), and z in (-infty,infty). They are related to Cartesian coordinates by

(x^2)/(a^2cosh^2u)+(y^2)/(a^2sinh^2u)=1
(4)
(x^2)/(a^2cos^2v)-(y^2)/(a^2sin^2v)=1.
(5)

The scale factors are

h_u=asqrt(cosh^2usin^2v+sinh^2ucos^2v)
(6)
=asqrt((cosh(2u)-cos(2v))/2)
(7)
=asqrt(sinh^2u+sin^2v)
(8)
h_v=asqrt(cosh^2usin^2v+sinh^2ucos^2v)
(9)
=asqrt((cosh(2u)-cos(2v))/2)
(10)
=asqrt(sinh^2u+sin^2v)
(11)
h_z=1.
(12)

The matrices of Christoffel symbols of the second kind in the sense of Misner et al. (1973) are given by

Gamma^u=[0 0 0; (sqrt(2)sin(2v))/(a[cosh(2u)-cos(2v)]^(3/2)) -(sqrt(2)sinh(2u))/(a[cosh(2u)-cos(2v)]^(3/2)) 0; 0 0 0]
(13)
Gamma^v=[-(sqrt(2)sin(2v))/(a[cosh(2u)-cos(2v)]^(3/2)) (sqrt(2)sinh(2u))/(a[cosh(2u)-cos(2v)]^(3/2)) 0; 0 0 0; 0 0 0]
(14)
Gamma^z=[0 0 0; 0 0 0; 0 0 0].
(15)

The Jacobian is

|(partial(x,y,z))/(partial(u,v,z))|=1/2a^2[cosh(2u)-cos(2v)].
(16)

The Laplacian is

del ^2=1/(a^2(sinh^2u+sin^2v))((partial^2)/(partialu^2)+(partial^2)/(partialv^2))+(partial^2)/(partialz^2).
(17)

Let

q_1=coshu
(18)
q_2=cosv
(19)
q_3=z.
(20)

Then the new scale factors are

h_(q_1)=asqrt((q_1^2-q_2^2)/(q_1^2-1))
(21)
h_(q_2)=asqrt((q_1^2-q_2^2)/(1-q_1^2))
(22)
h_(q_3)=1.
(23)

The Helmholtz differential equation is separable in elliptic cylindrical coordinates.

See also

Cylindrical Coordinates, Helmholtz Differential Equation--Elliptic Cylindrical Coordinates

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References

Arfken, G. "Elliptic Cylindrical Coordinates (u, v, z)." §2.7 in Mathematical Methods for Physicists, 2nd ed. Orlando, FL: Academic Press, pp. 95-97, 1970.Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco: W. H. Freeman, 1973.Moon, P. and Spencer, D. E. "Elliptic-Cylinder Coordinates (eta,psi,z)." Table 1.03 in Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, 2nd ed. New York: Springer-Verlag, pp. 17-20, 1988.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 657, 1953.

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Elliptic Cylindrical Coordinates

Cite this as:

Weisstein, Eric W. "Elliptic Cylindrical Coordinates." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticCylindricalCoordinates.html

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