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.

Two-channel Siegert pseudostate approach to a three-body Coulomb problem: S -wave resonances of Ps − ( N = 2 , 3 )

Abstract: The recently developed two-channel Siegert pseudostate method of Sitnikov and Tolstikhin [Phys. Rev. A 67, 032714 (2003)] is here applied to the case of ${\mathrm{Ps}}^{\ensuremath{-}}$ ${(e}^{+}ee)$ exploiting the hyperspherical elliptic coordinates method. S-wave resonances and antiresonances are calculated and classified according to their locations in the complex plane of the uniformization parameter. The resonance positions and widths are compared to other accurate calculations.

Stäckel separability for Newton systems of cofactor type

Abstract: A conservative Newton system ¨ q = −∇V (q) in R n is called separa- ble when the Hamilton-Jacobi equation for the natural Hamiltonian H = 1 p 2 +V (q) can be solved through separation of variables in some curvilinear coordinates. If these coordinates are orhogonal, the New- ton system admits n first integrals, which all have separable Stackel form with quadratic dependence on p. We study here separability of the more general class of Newton systems ¨ q = −(cof G) −1 ∇W(q) that admit n quadratic first integrals. We prove that a related system with the same integrals can be trans- formed through a non-canonical transformation into a Stackel separa- ble Hamiltonian system and solved by qudratures, providing a solution to the original system. The separation coordinates, which are defined as characteristic roots of a linear pencil G −µ ˜ G of elliptic coordinates matrices, gener- alize the well known elliptic and parabolic coordinates. Examples of such new coordinates in two and three dimensions are given. These results extend, in a new direction, the classical separability theory for natural Hamiltonians developed in the works of Jacobi, Liouville, Stackel, Levi-Civita, Eisenhart, Benenti, Kalnins and Miller.

Iterative solution to the tm scattering by two infinitely long lossy dielectric elliptic cylinders

Abstract: An analytic solution to the problem of a plane electromagnetic wave scattering by two infinitely long lossy dielectric elliptic cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders. To compute the higher order terms of the scattered fields, the translation addition theorem for Mathieu functions is implemented to express the field scattered by one lossy dielectric cylinder in terms of the elliptic coordinate system of the other cylinder in order to impose the boundary conditions. Scattered field coefficients of various orders are obtained and written in matrix form. Numerical results are obtained for the scattered field in the far zone for different axial ratios, electrical separations, relative dielectric constants, and angles of incidence.

On orthogonal curvilinear coordinate systems in constant curvature spaces

Abstract: We describe a method for constructing an n-orthogonal coordinate system in constant curvature spaces. The construction proposed is actually a modification of the Krichever method for producing an orthogonal coordinate system in the n-dimensional Euclidean space. To demonstrate how this method works, we construct some examples of orthogonal coordinate systems on the two-dimensional sphere and the hyperbolic plane, in the case when the spectral curve is reducible and all irreducible components are isomorphic to a complex projective line.

Stresses in Half-Elliptic Curved Beams Subjected to Transverse Tip Forces

Abstract: In-plane bending of curved beams produces substantial through-thickness normal and shear stresses that can result in structural failures. A half-elliptic curved beam, having a known prescribed variable radius of curvature, is studied as an extension of the previously published circular arc beam. The equations for the normal, tangential and shear stresses are developed for a curved beam outlined by two confocal half ellipses loaded by a pair of concentrated perpendicular forces on its ends. Closed-form analytical solutions for the stresses are found using an elasticity approach, where the solution is found by using selected terms of the bi-harmonic equation in elliptic coordinates. For the case of an elliptic beam with an aspect ratio of very close to unity, the solution closely agrees with published circular beam solutions. For other elliptic beam aspect ratios, the calculated stresses display good correlation to detailed finite element model solutions. A parametric study revealed that the maximum normal stress is located at the mid-plane for high-aspect ratio (a/b1) half-elliptic beams, but shifts toward the load tip for low aspect ratio (a/b1) beams due to local curvature effects. Moreover, the peak shear stress location moves toward the mid-plane and the magnitude greatly increases as the aspect ratio is increased. Thus, there are large normal and shear stress interactions occurring near the mid-plane for high-aspect ratio halfelliptic beams, which is not observed for circular beams. These stress interactions can produce unique failures in materials having low shear strength and through-thickness strength. The framework presented is valid for elliptic beams of any thickness or aspect ratios. The current closed-form solution is an improvement on previously published approximate solutions.