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.

Quantized Gravitational Field

Abstract: A gravitational action operator is constructed that is invariant under general coordinate transformations and local Lorentz (gauge) transformations To interpret the formalism the arbitrariness in description must be restricted by introducing gauge conditions and coordinate conditions The time gauge is defined by locking the time axes of the local coordinate systems to the general coordinate time axis The resulting form of the action operator, including the contribution of a spinless matter field, enables canonical pairs of variables to be identified There are four field variables that lack canonical partners, in virtue of differential constraint equations, which can be interpreted as space-time coordinate displacements In a physically distinguished class of coordinate system the gravitational field variables are not explicit functions of the coordinate displacement parameters There remains the freedom of Lorentz transformation The generators of spatial translations and rotations have the correct commutation properties The question of Lorentz invariance is left undecided since the energy density operator is only given implicitly

Diatomic Molecules According to the Wave Mechanics I: Electronic Levels of the Hydrogen Molecular Ion

Abstract: The electronic energies ${W}_{\ensuremath{\rho}}({n}_{y}, {n}_{\ensuremath{\varphi}}, {n}_{x})$ of the hydrogen molecular ion are calculated by means of the wave mechanics as functions of the nuclear separation $c=2\ensuremath{\rho}$, for several values of the quantum numbers ${n}_{y}$, ${n}_{\ensuremath{\varphi}}$ and ${n}_{x}$. The wave function is separable in the elliptical coordinates $y=\frac{({r}_{1}+{r}_{2})}{2\ensuremath{\rho}}$, $\ensuremath{\varphi}$ and $x=\frac{({r}_{1}\ensuremath{-}{r}_{2})}{2\ensuremath{\rho}}$. A qualitative idea of the behavior of these energies as $\ensuremath{\rho}$ changes from infinity to zero is gotten by an investigation of the behavior of the nodal surfaces. The number of these surfaces in any coordinate equals the quantum number in that coordinate. When $\ensuremath{\rho}=\ensuremath{\infty}$ the resulting system is that of a hydrogen atom and a separated nucleus, the nodes are paraboloids and planes with quantum numbers ${n}_{\ensuremath{\eta}}$, ${n}_{\ensuremath{\varphi}}$ and ${n}_{\ensuremath{\xi}}$, and the electronic energy is ${W}_{\ensuremath{\infty}}=\frac{R}{{({n}_{\ensuremath{\eta}}+{n}_{\ensuremath{\varphi}}+n\ensuremath{\xi}+1)}^{2}}$ where $R$ is the lowest energy of the hydrogen atom. When $\ensuremath{\rho}=0$ the system is that of a helium ion, the nodes are spherically symmetric with quantum numbers ${n}_{r}$, ${n}_{\ensuremath{\varphi}}$ and ${n}_{\ensuremath{\theta}}$, and the electronic energy is ${W}_{0}=\frac{4R}{{({n}_{r}+{n}_{\ensuremath{\varphi}}+{n}_{\ensuremath{\theta}}+1)}^{2}}$. As $\ensuremath{\rho}$ changes from zero to infinity it is shown that the quantum numbers are related in the manner ${n}_{r}\ensuremath{\rightarrow}{n}_{y}\ensuremath{\rightarrow}{n}_{\ensuremath{\eta}}$; ${n}_{\ensuremath{\varphi}}\ensuremath{\rightarrow}{n}_{\ensuremath{\varphi}}\ensuremath{\rightarrow}{n}_{\ensuremath{\varphi}}$; ${n}_{\ensuremath{\theta}}\ensuremath{\rightarrow}{n}_{x}\ensuremath{\rightarrow}2{n}_{\ensuremath{\xi}}$ or $2{n}_{\ensuremath{\xi}}+1$. Therefore ${W}_{0}=\frac{4R}{{{n}_{\ensuremath{\eta}}+{n}_{\ensuremath{\varphi}}+2{n}_{\ensuremath{\xi}}+1)}^{2}} or =\frac{4R}{{({n}_{\ensuremath{\eta}}+{n}_{\ensuremath{\varphi}}+2{n}_{\ensuremath{\xi}}+2)}^{2}}$. By this rule it is possible to check the following quantitative calculations. The first order perturbations of the various electronic energies of the first three degenerate levels of the helium ion resulting when $\ensuremath{\rho}=0$ were calculated; the perturbation being the slight separation of the nuclei ($\ensuremath{\rho}g0$). The first order perturbations of the various electronic energies of the first two degenerate levels of the hydrogen atom resulting when $\ensuremath{\rho}=\ensuremath{\infty}$ were calculated when the perturbation was the diminution of the separation ($\ensuremath{\rho}l\ensuremath{\infty}$). The first method is not valid for $\ensuremath{\rho}g\frac{a}{2}$, where $a$ is the radius of the first Bohr orbit of the hydrogen atom, and the second is not valid for $\ensuremath{\rho}l\frac{3a}{2}$. The gap between was extrapolated by means of the nodal reasoning above. These electronic energies plus the energy of nuclear repulsion give the molecular potential energies. A calculation of these shows that of the eight curves obtained only three, the $1s\ensuremath{\sigma}$, $3d\ensuremath{\sigma}$ and $4f\ensuremath{\sigma}$ states show minima, and therefore are stable configurations to this order of approximation (the Hund molecular notation is used for the states). The numerical results check with previous calculations and with the data available.

Polar coordinate based nonlinear function for frequency-domain blind source separation

Abstract: This paper presents a new type of nonlinear function for independent component analysis to process complex-valued signals, which is used in frequency-domain blind source separation. The new function is based on the polar coordinates of a complex number, whereas the conventional one is based on the Cartesian coordinates. The new function is derived from the probability density function of frequency-domain signals that are assumed to be independent of the phase. We show that the difference between the two types of functions is in the assumed densities of independent components. Experimental results for separating speech signals show that the new nonlinear function behaves better than the conventional one.

On the integrability and perturbation of three-dimensional fluid flows with symmetry

Abstract: The purpose of this paper is to develop analytical methods for studyingparticle paths in a class of three-dimensional incompressible fluid flows. In this paper we study three-dimensionalvolume preserving vector fields that are invariant under the action of a one-parameter symmetry group whose infinitesimal generator is autonomous and volume-preserving. We show that there exists a coordinate system in which the vector field assumes a simple form. In particular, the evolution of two of the coordinates is governed by a time-dependent, one-degree-of-freedom Hamiltonian system with the evolution of the remaining coordinate being governed by a first-order differential equation that depends only on the other two coordinates and time. The new coordinates depend only on the symmetry group of the vector field. Therefore they arefield-independent. The coordinate transformation is constructive. If the vector field is time-independent, then it possesses an integral of motion. Moreover, we show that the system can be further reduced toaction-angle-angle coordinates. These are analogous to the familiar action-angle variables from Hamiltonian mechanics and are quite useful for perturbative studies of the class of systems we consider. In fact, we show how our coordinate transformation puts us in a position to apply recent extensions of the Kolmogorov-Arnold-Moser (KAM) theorem for three-dimensional, volume-preserving maps as well as three-dimensional versions of Melnikov's method. We discuss the integrability of the class of flows considered, and draw an analogy with Clebsch variables in fluid mechanics.