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.

Riemannian Acceleration in Oblate Spheroidal Coordinate System

Abstract: The planetary bodies are more of a spheroid than they are a sphere thereby making it necessary to describe motions in a spheroidal coordinate system. Using the oblate spheroidal coordinate system, a more approximate and realistic description of motion in these bodies can be realized. In this paper, we derive the Riemannian acceleration for motion in oblate spheroidal coordinate system using the golden metric tensor in oblate spheroidal coordinates. The Riemannian acceleration in the oblate spheroidal coordinate system reduces to the pure Newtonian acceleration in the limit of c0 and contains post-Newtonian correction terms of all orders of c-2. The result obtained thereby opens the way for further studies and applications of the motion of particles in oblate spheroidal coordinate system.

Two-Dimensional Scattering by a Conducting Elliptic Cylinder Coated With a

Abstract: A solution to the two-dimensional scattering proper- ties of a conducting elliptic cylinder coated with a confocal ho- mogeneous 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 direc- tions 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 proper- ties of Mathieu functions, Galerkin's method is applied and a ma- trix is required for the computation of unknown expansion coeffi- cients of the scattered and transmitted fields. Only the transverse magnetic (TM) polarization is presented, while the transverse elec- tric (TE) polarization can be obtained in the same way. Some nu- merical 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.

On derivations of basic governing equations for solid bodies in bispherical coordinate system

Abstract: Abstract Traditionally solid mechanics problems are formulated in Cartesian, cylindrical and spherical coordinate system. Using such formulation and coordinate system, solutions of solid mechanics problems are obtained for specific geometries such as straight boundary, circular, cylindrical and spherical boundaries. However, such available coordinate system cannot describe many geometries in spherical or cylindrical coordinate system with inclusion of eccentricities, two spherical or cylindrical bodies in contact and parallel. In this article, author address this issue by giving complete original formulations and derivations of basic governing partial differential equations in Bispherical coordinate system. Bispherical coordinate system allow us to solve problems such as eccentric spheres, two parallel spheres (intersecting or non-intersecting) in solid mechanics which traditional spherical coordinate system cannot handle. Author develop the original equations in bispherical coordinates in terms of useful quantities of stresses and strains components in three dimensions. The paper is limited to basic formulations and practical applications of such formulations can easily be expanded for getting solutions using any known analytical or numerical methods.

Hitting distribution of a correlated planar Brownian motion in a disk

Abstract: In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $\rho$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $\rho$ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.