↑ arvau

Wikipedia / Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system-direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci-axis of the Cartesian coordinate system.

Basic definition

is

.

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

form ellipses, whereas the hyperbolic trigonometric identity

form hyperbolae.

Scale factors

are equal

. Consequently, an infinitesimal volume element equals

and the Laplacian equals

by substituting the scale factors into the general formulae found in orthogonal coordinates.

Alternative definition

must belong to the interval coordinate must be greater than or equal to one.

. For any point in the (x,y) plane, the sum, whereas their difference, respectively.)

A drawback of these coordinates is that they do not have a 1-to-1 transformation to the Cartesian coordinates

Alternative scale factors

are

. Hence, the infinitesimal volume element becomes

and the Laplacian equals

by substituting the scale factors into the general formulae found in orthogonal coordinates.

Applications

The classic applications of elliptic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which elliptic cylindrical coordinates allow a separation of variables. A typical example would be the electric field.

The three-dimensional wave equation, when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equations.

could represent the momenta of a particle and its decomposition products, respectively, and the integrand might involve the kinetic energies of the products (which are proportional to the squared lengths of the momenta).

Bibliography

  • Same as Morse & Feshbach (1953), substituting uk for ξk.