Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

Separable Potentials and a Triality in Two-Dimensional Spaces of Constant Curvature

Abstract: We characterize and completely describe some types of separable potentials in twodimensional spaces, S 2 [�1]�2 , of any (positive, zero or negative) constant curvature and either definite or indefinite signature type. The results are formulated in a way which applies at once for the two-dimensional sphere S 2 , hyperbolic plane H 2 , AntiDeSitter / DeSitter two-dimensional spaces AdS 1+1 / dS 1+1 as well as for their flat analogues E 2 and M 1+1 . This is achieved through an approach of Cayley-Klein type with two parameters, �1 and �2, to encompass all curvatures and signature types. We discuss six coordinate systems allowing separation of the Hamilton-Jacobi equation for natural Hamiltonians in S 2�1]�2 and relate them by a formal triality transformation, which seems to be a clue to introduce general “elliptic coordinates” for any CK space concisely. As an application we give, in any S 2 [�1]�2 , the explicit expressions for the Fradkin tensor and for the Runge-Lenz vector, i.e., the constants of motion for the harmonic oscillator and Kepler potential on any S 2

Two-Dimensional Scattering by a Conducting Elliptic Cylinder Coated With a Homogeneous Anisotropic Shell

Abstract: A solution to the two-dimensional scattering properties of a conducting elliptic cylinder coated with a confocal homogeneous anisotropic elliptical shell is obtained. The transmitted field of the anisotropic shell is expressed as an integral equation based on waves with different wave numbers and different directions of propagation. The waves in all directions are represented as the eigenfunction expansion in elliptic coordinates in terms of Mathieu functions. In order to solve the nonorthogonality properties of Mathieu functions, Galerkin's method is applied and a matrix is required for the computation of unknown expansion coefficients of the scattered and transmitted fields. Only the transverse magnetic (TM) polarization is presented, while the transverse electric (TE) polarization can be obtained in the same way. Some numerical results are presented in graphical forms. The result is in agreement with that available as expected when a coated elliptic cylinder degenerates to the coated circular one.

Applications of fractional exterior differential in three-dimensional space

Abstract: A brief survey of fractional calculus and fractional differential forms was firstly given The fractional exterior transition to curvilinear coordinate at the origin were discussed and the two coordinate transformations for the fractional differentials for three-dimensional Cartesian coordinates to spherical and cylindrical coordinates are obtained, respectively In particular, for v=m=1, the usual exterior transformations, between the spherical coordinate and Cartesian coordinate, as well as the cylindrical coordinate and Cartesian coordinate, are found respectively, from fractional exterior transformation