Showing papers on "Elliptic coordinate system published in 2006"

Solving Einstein's equations with dual coordinate frames

Abstract: A method is introduced for solving Einstein's equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the way fields are projected onto an orthonormal tetrad basis. These field components are then determined as functions of a second independent coordinate system. The transformation to the second coordinate system can be thought of as a mapping from the original inertial coordinate system to the computational domain. This dual-coordinate method is used to perform stable numerical evolutions of a black-hole spacetime using the generalized harmonic form of Einstein's equations in coordinates that rotate with respect to the inertial frame at infinity; such evolutions are found to be generically unstable using a single rotating-coordinate frame. The dual-coordinate method is also used here to evolve binary black-hole spacetimes for several orbits. The great flexibility of this method allows comoving coordinates to be adjusted with a feedback control system that keeps the excision boundaries of the holes within their respective apparent horizons.

Generation of an elliptic Bessel beam

Abstract: In an elliptic coordinate system the solution for a diffraction-free beam is given by a Mathieu beam, which is mathematically complicated and therefore takes considerable computational time and a large memory space. A technique is reported for the generation of elliptic hollow beams, which are actually scaled versions of higher-order Bessel beams. The analysis is based on the fact that stretching coordinates in the space domain results in a contraction of the coordinates in the frequency domain, along with a change in the overall amplitude of the spectrum. The beam, produced holographically, is structurally stable and retains its shape up to approximately 1 m.

Notes on the Dipole Coordinate System

Abstract: A strong magnetic field can make it advantageous to work in a coordinate system aligned with dipolar field lines. This monograph collect the formulas for some of the most frequently used expressions and operations in dipole coordinates.

Natural Steady Convection in a Space Annulus Between Two Elliptic Confocal Ducts: Influence of the Slope Angle

Abstract: The authors express the Boussinesq equations of the laminar thermal and natural convection, in the case of permanent and bidimensional flow, in an annular space between two confocal elliptic cylinders. The latter is oriented at an arbitrary angle a with respect to the gravity force, using the elliptic coordinates system. A new calculation code using the finite volumes with the primitive functions (velocity-pressure formulation) is proposed. The Prandtl number is fixed at 0.7 (case of the air) with varying the Rayleigh number. The effect of the system inclination is examined.

Modal expansion for the 2D Green's function in a non-orthogonal coordinates system.

Abstract: We present an efficient modal method to calculate the two-dimensional Green's function for electromagnetics in curvilinear coordinates. For this purpose the coordinate transformation based differential method, introduced for the numerical analysis of surface-relief gratings, is directly used with perfectly matched layers (PMLs). The covariant formalism Maxwell's equations, very convenient for the non-orthogonal coordinates formulation, also gives an unified analysis of PMLs. Numerical results for a line source placed above a perfectly conducting corrugated surface are presented.

A note on elliptic coordinates on the Lie algebra e(3)

Abstract: We introduce elliptic coordinates on the dual space to the Lie algebra e(3) and discuss the separability of the Clebsch system in these variables. The proposed Darboux coordinates on e*(3) coincide with the usual elliptic coordinates on the cotangent bundle of the two-dimensional sphere at the zero value of the corresponding Casimir function.

Natural Coordinate System in Curved Space-time

Abstract: In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be preseted according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it "natural coordinate system". The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In NCS we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein's equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is a nice coordinate system to describe the realistic curved space-time, and is helpful to understand the nature of space-time.

Transverse Electric Plane Wave Scattering by Two Infinitely Long Conducting Elliptic Cylinders: Iterative Solution

Abstract: An analytic solution to the problem of a TE polarized plane electromagnetic wave scattering by two infinitely long conducting 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 cylinder in terms of the elliptic coordinate system of the other cylinder in order to impose the boundary conditions. Scattered field coefficients of various scattered orders are obtained and written in matrix form. Numerical results are obtained for the scattered field in the far zone for different axial ratio, electrical separation distance and angles of incidence.

Some aspects of theoretical analysis of the light propagation in an elliptical optical fiber

Abstract: The paper presents the results of theoretical analysis of the propagation of the electromagnetic field in optical fiber with an elliptic core. Due to the elliptic symmetry of the core the analysis had to be carried out in an elliptic coordinate system. This analysis has shown that the functions describing the z-component of the electric field satisfy the Mathieu equations. The presented results of the theoretical analysis are original and have not published so for.

Reductions to Korteweg-de Vries Soliton Hierarchy

Abstract: Based on the nonlinearization of Lax pairs, the Korteweg-de Vries (KdV) soliton hierarchy is decomposed into a family of finite-dimensional Hamiltonian systems, whose Liouville integrability is proved by means of the elliptic coordinates. By applying the Abel–Jacobi coordinates on a Riemann surface of hyperelliptic curve, the resulting Hamiltonian flows as well as the KdV soliton hierarchy are ultimately reduced into linear superpositions, expressed by the Abel–Jacobi variables.