Showing papers on "Elliptic coordinate system published in 1973"

Solutions of plane crack problems by mapping technique

Abstract: The analysis of crack problems in plane elasticity has intrigued mathematicians for nearly sixty years. Inglis [1] found the solution for a single crack in an infinite sheet with the use of elliptic coordinates. Since then, many mathematical approaches with wide ranges of sophistication have been applied to a variety of crack configurations and loading conditions. It is easy to appreciate the mathematical interest in a problem area in which solution techniques span such diverse topics as analytic function theory, integral equations, transform methods, conformal mapping, boundary collocation, finite differences, finite elements, asymptotic methods, etc.

Scalar waves in the mixmaster universe. I. The Helmholtz equation in a fixed background

Abstract: Solutions to the scalar wave equation in a fixed mixmaster background are presented. Separability conditions on the Laplace-Beltrami equation are related to the complete sets of invariant operators of the symmetry group S${\mathrm{O}}_{3}$. Solutions to the Helmholtz equation are equivalent to the quantum-mechanical problem of asymmetric rotors. The wave functions in the mixmaster space are the asymmetric-top wave functions. In Euler angle variables, they are expanded in terms of the symmetric-top wave functions (the wave functions in Taub space) possessing definite ($J, K, M$) symmetries, with a coupling in the intrinsic magnetic quantum number $K$; the case with $M=0$ can be expressed as a product of the Lam\'e functions in elliptic coordinates. Invariance of the mixmaster space under the four - group characterizes the wave functions into four symmetry species and causes the factorization of the energy matrix into four submatrices. Eigenvalues and expansion coefficients for the wave functions are calculated for some low-lying levels.

Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system

Abstract: Much interest has developed in recent years in the flow of non-Newtonian fluids and many excellent articles and books have appeared upon this subject (1, 2, 3, 4). However, most of the solutions which have been published have been for steady flows. In this paper the nonlinear equations of motion in general orthogonal coordinates are developed for the non-steady flow of an incompressible non-Newtonian fluid. A solution is presented for the non-steady Hagen-Poiseuille flow through a circular pipe.

Variational approximations to the 2pσ u and 2pπ u wavefunctions of the hydrogen molecule ion

Abstract: It is shown that trial wavefunctions for the 2p sigma u and 2p pi u states of H2+ with three variational parameters, separable in elliptic coordinates give excellent results for the energy eigenvalues as functions of the internuclear coordinate. It is also seen that there are two minima for the energy as a function of the parameters. Earlier results for the same trial function used for the 2p sigma u state in error because the optimization has not always been carried out for the absolute minimum.

An improved coordinate transformation for the treatment of a paramagnetic atom in a linear oscillatory and a constant transverse magnetic field

Abstract: The introduction of a suitably moving coordinate system results in approximate equations of motion which lead in a direct manner to the Block-Siegert resonance shift, to 2nd or higher order, and to a simple visualization of multiple quantum transitions, including their correct quantitative treatment. The relation between the present method and that due to Shirley (1965), whose results are confirmed, is pointed out.

A Coordinate System Based on a Twisting Null Geodesic Congruence

Abstract: A coordinate system based on a twisting null geodesic congruence is developed. The various freedoms of choice are investigated and a short theorem regarding the choice of one coordinate surface is proved.