Concircular tensors in Spaces of Constant Curvature: With Applications to Orthogonal Separation of The Hamilton-Jacobi Equation

We study concircular tensors in spaces of constant curvature and then apply the results obtained to the problem of the orthogonal separation of the Hamilton-Jacobi equation on these spaces. Any coordinates which separate the geodesic Hamilton-Jacobi equation are called separable. Specifically for spaces of constant curvature, we obtain canonical forms of concircular tensors modulo the action of the isometry group, we obtain the separable coordinates induced by irreducible concircular tensors, and we obtain warped products adapted to reducible concircular tensors. Using these results, we show how to enumerate the isometrically inequivalent orthogonal separable coordinates, construct the transformation from separable to Cartesian coordinates, and execute the Benenti-Eisenhart-Kalnins-Miller (BEKM) separation algorithm for separating natural Hamilton-Jacobi equations.Comment: Removed preamble and references to unpublished articles. Also made some minor changes in the bod

Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory revolves around a special type of conformal Killing tensor, hereafter called a concircular tensor. First, we show how to extend original results given by Benenti to intrinsically characterize all (orthogonal) separable coordinates in spaces of constant curvature using concircular tensors. This results in the construction of a special class of separable coordinates known as Kalnins-Eisenhart-Miller coordinates. Then we present the Benenti-Eisenhart-Kalnins-Miller separation algorithm, which uses concircular tensors to intrinsically search for Kalnins-Eisenhart-Miller coordinates which separate a given natural Hamilton-Jacobi equation. As a new application of the theory, we show how to obtain the separable coordinate systems in the two dimensional spaces of constant curvature, Minkowski and (Anti-)de Sitter space. We also apply the Benenti-Eisenhart-Kalnins-Miller separation algorithm to study the separability of the three dimensional Calogero-Moser and Morosi-Tondo systems

Bubble kinematics in a sheared foam

We characterize the kinematics of bubbles in a sheared two-dimensional foam using statistical measures. We consider the distributions of both bubble velocities and displacements. The results are discussed in the context of the expected behavior for a thermal system and simulations of the bubble model. There is general agreement between the experiments and the simulation, but notable differences in the velocity distributions point to interesting elements of the sheared foam not captured by prevalent models

Circularly Polarized Antenna with Metallic Reflector for High-Gain Satellite Communication

This paper describes the design and performance of a circularly polarized antenna for satellite communication applications. The antenna design includes a monopole patch antenna, split ring resonator, defective ground surface and a metallic reflector. The antenna is designed to operate in multiple frequency bands and to have high gain. The metallic reflector is used to enhance the antenna gain by reflecting the signal in the same phase as the transmitted signal, thereby strengthening the signal. The antenna resonates in 4.4 GHz and 7.6 GHz frequency bands, making it suitable for satellite communication applications. The paper presents both simulated and measured results of the antenna's performance, including return loss, gain, VSWR and polarization characteristics. The paper provides a detailed analysis of antenna's performance, demonstrating its suitability for satellite communication applications. The use of a metallic reflector, the split ring resonator and defective ground surface design results into antenna's high gain and multiband operation

Ionospheric flow shear associated with the preexisting auroral arc: A statistical study from the FAST spacecraft data

An auroral substorm is a disturbance in the magnetosphere that releases energy stored in the magnetotail into the high‐latitude ionosphere. By definition, an auroral substorm commences when a discrete auroral arc brightens and subsequently expands poleward and azimuthally. The arc that brightens is usually the most equatorward of several auroral arcs that remain quiescent for ~5 to ~60 min before the breakup commences. This arc is often referred to as the “preexisting auroral arc (PAA)” or the “growth‐phase arc.” In this study, we use FAST measurements to establish the statistics of flow patterns near PAAs in the ionosphere. We find that flow shear is present in the vicinity of a preexisting arc. When a PAA appears in the evening sector, enhanced westward flow develops equatorward of the arc, whereas when a PAA appears in the morning sector, enhanced eastward flow develops poleward of the arc. We benchmark locations of the PAAs relative to large‐scale field‐aligned currents (FACs) and convective flows in the ionosphere, finding that the arc forms in the upward current region within ~1° of the Region 1/Region 2 boundary in all local time sectors from 20 MLT to 03 MLT. We also find that near midnight in the Harang region, most of the PAAs lie within 0.5° poleward of the low‐latitude Region 1/Region 2 currents boundary and sit between the westward and eastward flow peak but equatorward of the flow reversal point. Finally, we examine arc‐associated electrodynamics and find that the FAC of the PAA is mainly closed by the north‐south Pedersen current in the ionosphere.Key PointsAn ionospheric flow shear is associated with the preexisting auroral arcThe FAC of the PAA is primarily closed by N‐S Pedersen current in the ionosphereThe PAA is located very close to the R1/R2 boundaryPeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/112278/1/jgra51768.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/112278/2/jgra51768-sup-0001-supinfo.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/112278/3/jgra51768-sup-0002-supinfo.pd