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.

A Comparative Study of Modeling the Magnetostatic Field in a Current-Carrying Plate Containing an Elliptic Hole

Abstract: The presence of crack-like defects can cause an uneven distribution of the electric current density in a cracked conductor. To investigate the perturbation of the magnetic field resulting from the disturbed electric current, computational modeling of the magnetostatics is attempted on an infinite conductive plate, which contains an elliptic hole and is subjected to uniform current flow at infinity. Both 2-D and 3-D analyses are considered in this study. The 2-D analysis requires certain crucial assumptions and the governing Maxwell's equations are solved analytically in elliptic coordinates. The 3-D numerical computation is based on superposition of the elementary solution, whose derivation utilizes the Biot-Savart law. To improve the efficiency of the 3-D calculation, an adaptive mesh refinement algorithm is implemented in the numerical discretization. Finally, through a comparative study, the validity of the introduced simplifications in the 2-D analysis is benchmarked with the 3-D computational results. The present study shows that the 2-D solution predicts the upper bound for the out-of-plane component of the magnetic field perturbed by the elliptical hole, whose semi-major axis does not exceed ten times the thickness of the plate.

A solution of the motion of a charged particle in magnetic mirror systems

Abstract: The paper solves the motion of a charged particle in an axially symmetrical, magnetostatic field, which forms magnetic mirror systems. The solution is based on the Hamiltonian for the motion of a charged particle in such a field. The whole problem is solved in orthogonal curvilinear coordinates, the coordinate curves being formed by the lines of force and the curves perpendicular to them. By applying the perturbation method of multi-periodical systems to the Hamiltonian in these coordinates, a Hamiltonian is obtained for the motion of the guiding centre which contains only the coordinate of motion along the lines of force, the others being cyclic. Thus, in addition to the energy two other first integrals of the equations of motion are obtained (since the impulses corresponding to the cyclic coordinates are constant), from which the conditions for confinement of motion in a magnetic field are immediately obtained. Since the Hamiltonian obtained in this way contains only one coordinate, the whole problem is solved by two quadratures, which define the dependence of the time and azimuthal angle on the line-of-force coordinate, while the other quantities are constant.

Calibration of a multi-plane X-ray unit

Abstract: For calibrating a multi-plane X-ray unit, an internal matrix mapping coordinate system internal to the unit onto coordinate system of a first radiographic plane is predefined in a reference projection geometry. An external matrix mapping coordinate system external to the unit onto coordinate system of the first plane is determined from image data captured by the first plane at calibration points and coordinates of the calibration points. An external matrix mapping coordinate system external to the unit onto coordinate system of a second radiographic plane is determined from image data captured by the second plane at the calibration points and the coordinates of the calibration points. A measure of position of the second plane with respect to coordinate system internal to the unit or with respect to the first plane is determined from the internal and external matrix of the first plane and the external matrix of the second plane.

Screening effects on the electronic structure of the hydrogen molecular ion

Abstract: We study the effect that a statically screened Coulomb potential represented by a Debye-H\"uckel-Yukawa potential has in the electronic structure of the simplest molecule ${{\mathrm{H}}_{2}}^{+}$ within the Born-Oppenheimer approximation. The method of solution is based on a two-center partial-wave expansion expressed in confocal elliptic coordinates using B-spline polynomials. General algorithms for the computation of energies, wave functions, and dipole and nonadiabatic radial matrix elements are given in detail. As it occurs in atoms, screening in simple molecules shifts the energies of bound states upwards so that, as screening increases, every bound state eventually crosses the upper ionization threshold at a critical screening value. The loss of long-range Coulomb interactions has its effect in the structure of wave functions, and consequently in the dipole and nonadiabatic matrix elements at intermediate and long internuclear distances, which determine the dynamics in external electromagnetic fields and collisional processes. Other issues related to a practical solution of the arbitrary sign problem, as well as the assignment of angular and radial nodes to the variational eigenfunctions, and the appearance of molecular shape resonances and Borromean states in ${{\mathrm{H}}_{2}}^{+}$ as screening increases, are also addressed in this work.