Topic
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.
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TL;DR: In the course of checking their work on the symbolic calculation of molecular integrals over Slater orbitals, this paper obtained some results in substantial disagreement with two recent articles that describe numerical schemes.
Abstract: In the course of checking our work on the symbolic calculation of molecular integrals over Slater orbitals, we obtained some results in substantial disagreement with two recent articles that describe numerical schemes. We believe that these schemes suffer from digital erosion, possibly because recurrence formulas were used outside their regions of stability. Our results were obtained using the ζ-function method, which expands the orbital on one atom onto the other, and integrates in polar coordinates. They were checked using elliptic coordinates. Both sets of calculations were performed symbolically. We summarize these calculations and discuss the impact of symbolic calculation on the accuracy of molecular computations.
35 citations
TL;DR: In this article, an orthogonal coordinate system for an arbitrary smooth curve was derived from the Serret-Frenet frame of the curve, which can be used for the study of wave propagation along curved structures, such as curved waveguides and antennas.
Abstract: To facilitate the study of wave propagation along curved structures, such as curved waveguides and antennas, it is desirable to have an orthogonal coordinate system for an arbitrary smooth curve. Such a coordinate system can be derived from the Serret-Frenet frame of the curve.
35 citations
35 citations
TL;DR: In this article, a complete classification of all orthogonal coordinate systems that admit a separation of variables for the null Hamilton-Jacobi equation in conformally flat complex Riemannian spaces is presented.
Abstract: A complete classification of all orthogonal coordinate systems that admit a separation of variables for the null Hamilton-Jacobi equation in conformally flat complex Riemannian spaces is presented. This is a first step towards the complete solution of the problem for complex Riemannian spaces when, in general, the coordinates need not be orthogonal. A detailed prescription for constructing all such orthogonal coordinate systems is presented.
35 citations
TL;DR: In this paper, it was shown that the Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation.
Abstract: The Neumann and Chaplygin systems on the sphere are simultaneously separable in variables obtained from the standard elliptic coordinates by the proper Backlund transformation. We also prove that after similar Backlund transformations other curvilinear coordinates on the sphere and on the plane become variables of separation for the system with quartic potential, for the Henon-Heiles system and for the Kowalevski top. This allows us to speak about some analog of the hetero Backlund transformations relating different Hamilton-Jacobi equations.
34 citations