Showing papers on "Elliptic coordinate system published in 2000"

Developments in determining the gravitational potential using toroidal functions

Abstract: Cohl & Tohline (1999) have shown how the integration/summation expression for the Green’s function in cylindrical coordinates can be written as an azimuthal Fourier series expansion, with toroidal functions as expansion coefficients. In this paper, we show how this compact representation can be extended to other rotationally invariant coordinate systems which are known to admit separable solutions for Laplace’s equation.

Potential energy surfaces and coordinate dependence

Abstract: For coordinate systems with a nontrivial metric tensor covariant derivatives must be used to obtain properties that are coordinate independent. Applications to instantaneous normal mode theory and bifurcation points are presented as illustrations.

On the stretching of Maxwell's equations in general orthogonal coordinate systems and the perfectly matched layer

Abstract: It is shown that the complex coordinate stretching and diagonal anisotropy formulations of the perfectly matched layer are equivalent in a general orthogonal coordinate system setting. The results are obtained by taking advantage of the tensorial invariance of the line element.

Full modal analysis of confocal coaxial elliptical waveguides

Abstract: An efficient method for analysing confocal coaxial elliptical waveguides is presented. Using elliptical coordinates, the differential Helmholtz equation is transformed into a linear matrix eigenvalue problem by means of the method of moments. The expressions of the vector mode functions for the full spectrum of these guides are constructed, including the TEM, TM and TE modes. The convergence of the method is very good, giving an efficient and accurate code. Comparisons with numerical results found in the technical literature validate the presented theory.

Boundary-condition-determined wave functions for H 2 + and its isoelectronic ions

Abstract: We have discussed some general properties of the wave functions of ${\mathrm{H}}_{2}^{+}$ and its isoelectronic ions, in terms of confocal elliptic coordinates. Based on the asymptotic properties when the electron is far away from the nuclei, and the coalescence properties when it is close to one of the nuclei, we have developed simple, model wave functions for several states. These wave functions are simple enough to provide a clear insight into the structure of the wave functions, and yet accurate enough to provide energies close to the exact values.

Applications of the solution to the Rayleigh problem

Abstract: A formalism to determine the probability density function (PDF) resulting from a coordinate transformation applied to an arbitrary PDF is developed. The results are applied to the coordinate transformation used in target tracking, specifically the transformation from spherical to Cartesian coordinates. This result is then applied to the specific example of a Gaussian random variable transformed by sine and cosine coordinate transformations. A brief discussion is then carried out of multi-dimensional transformations.

A computer algebra-based coordinate reduction technique for multi-body systems

Abstract: This paper presents a heuristic algorithm to solve the linear coordinate reduction problem. If Cartesian coordinates are chosen in the initial formulation the algorithm eliminates two-thirds of dependent coordinates in the planar case and one-half in the spatial case for mechanisms composed of spherical, revolute and universal joints. For an open-loop system composed of spherical joints it eliminates all dependent coordinates. A computer algebra-based implementation in the Maple language is presented. The proposed technique is demonstrated by application to the dynamic analysis of a Peaucellier mechanism.

Normal coordinate systems from a viewpoint of real analysis

Abstract: Normal coordinate systems for pseudo-Riemannian metrics are investigated from a viewpoint of the theory of partial differential equations. Given a cartesian coordinate system $x$, a local metric for which $x$ is a normal coordinate system is determined by a metric tensor at the origin and any one of certain three matrix functions. These are related one another by three partial differential equations. Solvability of these equations in $C^{\infty}$ framework and power series expansion of solutions are discussed.