On Backlund transformation and motion of null Cartan curves

The main scope of this paper is to examine null Cartan curves especially the ones with constant torsion. In accordance with this scope, the position vector of a null Cartan curve is stated by a linear combination of the vector fields of its pseudo-orthogonal frame with differentiable functions. However, the most important difference that distinguishes this study from the other studies is that the Bertrand curve couples (timelike, spacelike or null) of null Cartan curves are also examined. Consequently, it is seen that all kinds of Bertrand couples of a given null Cartan curve with constant curvature functions have also constant curvature functions. This result is the most valuable result of the study, but allows us to introduce a transformation on null Cartan curves. Then, it is proved that aforesaid transformation is a Backlund transformation which is well recognized in modern physics. Moreover, motion of an inextensible null Cartan curve is investigated. By considering time evolution of null Cartan curve, the angular momentum vector is examined. And three different situations are given depending on the character of the angular momentum vector [Formula: see text] In the case of [Formula: see text] we discuss the solution of the system which is obtained by compatibility conditions. Finally, we provide the relation between torsion of the curve and the velocity vector components of the moving curve [Formula: see text]

Evolution of the rotational motion of a viscoelastic planet with a core on an elliptical orbit

The work is devoted to the study of the evolution of the rotational motion of a planet in the central Newtonian field of forces. The planet is modeled by a body consisting of a solid core and a viscoelastic shell rigidly attached to it. A limited formulation of the problem is considered, when the center of mass of the planet moves along a given Keplerian elliptical orbit. The equations of motion are derived in the form of a system of Routh equations using the canonical Andoyer variables, which are “action-angle” variables in the unperturbed problem and have the form of integro-differential equations with partial derivatives. The technique developed by V.G. Vilke is used for mechanical systems with an infinite number of degrees of freedom. A system of ordinary differential equations is obtained by the method of separation of motions. The system describes the rotational motion of the planet taking into account the perturbations caused by elasticity and dissipation. An evolutionary system of equations for the “action” variables and slow angular variables is obtained by the averaging method. A phase portrait is constructed that describes the mutual change in the modulus of the angular momentum vector G of the rotational motion and the cosine of the angle between this vector and the normal to the orbital plane of the planet’s center of mass. A stationary solution of the evolutionary system of equations is found, which is asymptotically stable. It is shown that in stationary motion, the angular momentum vector G is orthogonal to the orbital plane, and the limiting value of the modulus of this vector depends on the eccentricity of the elliptical orbit. The constructed mathematical model can be used to study the tidal evolution of the rotational motion of planets and satellites. The results obtained in this work are consistent with the results of previous studies in this area.

Study of Spatial Orientation of Angular Momentum of z-Magnitude SDSS DR-13 Galaxies with Red Shift 0.50 to 0.53

The spatial orientations of 142,929 SDSS DR-13, z- magnitude galaxies having red shift 0.50 to 0.53 have been analyzed. The main goal of this work is to examine the orientation of the angular momentum of galaxies within the given redshift limit in the framework of three different scenarios 'Hierarchy model', 'Pancake model', and the 'Primordial vorticity model'. By using Godlowskian transformation the two-dimensional data were converted into three-dimensional data (polar and azimuthal angles). The expected isotropy distribution curves were obtained by removing the selection effects and performing a random simulation to generate 107 virtual galaxies by using Matlab 2015a. Three statistical tests of Chi-square, autocorrelation, and Fourier were used to compare the expected isotropic data with observed. The data classified into nine subsamples having each of one magnitude size. In general, the results supported the Hierarchy model. The model advocates random orientations of angular. However, a local anisotropy observed in few subsamples suggested a gravitational tidal interaction between neighboring galaxies, an early-merging process in which the angular momentum vector distorts the initial alignment of nearby galaxies.

Angular Momentum and the Quantum Gyroscope: The Emergence of Spin

Continuing with understanding the implications of the postulates in Chapter 7 and following the approach in Chapter 8 to use operators find solutions to the time independent Schrödinger equation, we return to the subject of angular momentum, of importance to many problems including the quantum gyroscope. Aside from playing a central role in any spherically symmetric quantum system, it plays a central role in inertial guidance systems from airplanes and rockets to autonomous vehicles. Working with only the operators of the angular momentum vector, L^=L^xx̌+L^yy̌+L^zž and L^2 and the corresponding commutation relations, a procedure similar to that used in Chapter 8 for the nano-vibrator is used to completely identify the eigenvectors and eigenvalues. However, in Chapter 6, we required that the magnetic quantum number, m where L^z|l.m〉=mℏ|l.m〉, be integer, because the eigenfunction Yl,m(l,m)∝eimϕ, and we required that a full rotation around the z-axis give the same result requiring, eimϕ=eim(ϕ+2π). In the operator approach, there is no such requirement, but there is still a constraint on m, namely that m is either integer or half integer. The requirements in Chapter 6 hold, so what is the meaning of half-integer? This was one of the first results to indicate the existence of intrinsic (not associated with real space rotation) angular moment known as spin.

Angular Momentum and Rotational Motion in Three Dimensions

This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating particles as the linear momentum 3-vector plays in the kinetic energy for translational motion. The quantum operator for the three components do not commute, i.e. the full vector cannot be known. Each component commutes with the operator for the square-length of the angular momentum vector, and therefore one opts for calculating the square-length and one of the components, z, referred to as the projection. The angular momentum is quantized. The (azimuthal) quantum number ℓ = 0,1,2,3… quantifies the square-length as ℓ(ℓ+1), and the projection is quantified by the (magnbetic) quantum number mℓ = -ℓ…ℓ in integer steps. The eigenfunctions are called spherical harmonics. A ‘rigid rotor’ model is set up to treat the rotational spectrum of a diatomic molecule.

Satellite Attitude Control Using Control Moment Gyroscopes

In space missions, there is often a need for an attitude control system capable of maintaining the desired attitude. In situations that require agile and accurate responses, which also require large torques, control moment gyroscopes (CMGs) may be used. Control moment gyroscopes are high angular moment gyros mounted on gimbals and are responsible for changing the direction of the angular momentum vector, consequently generating the control torques. There are several linear and nonlinear techniques that can be employed in the design of control laws with the final choice being a compromise between simplicity, effectiveness, efficiency and robustness. The main objective of this study is to evaluate the performance of control systems techniques with 4 CMGs in a pyramidal arrangement, either by using Linear Quadratic Tracker (LQT) with integral compensator or Exponential Mapping Control (EMC). A reference attitude will be defined to be traced in the presence of disturbance torques caused by the gravitational gradient.

Inversion of the longitudinal component of spin angular momentum in the focus of a left-handed circularly polarized beam

It has been shown theoretically and numerically that in the sharp focus of a circularly polarized optical vortex, the longitudinal component of the spin angular momentum vector is inverted. Moreover, if the input light to the optical system is left-hand circularly polarized, it has been shown to be right-hand polarized in the focus near the optical axis. Since this effect occurs near the focus where a backward energy flow takes place, such an inversion of the spin angular momentum can be used to detect the backward energy flow.

Deciphering the Lyman α blob 1 with deep MUSE observations

Context. Lyman α blobs (LABs) are large-scale radio-quiet Lyman α (Lyα) nebula at high-z that occur predominantly in overdense proto-cluster regions. In particular, there is the prototypical SSA22a-LAB1 at z = 3.1, which has become an observational reference for LABs across the electromagnetic spectrum. Aims. We want to understand the powering mechanisms that drive the LAB so that we may gain empirical insights into the galaxy-formation processes within a rare dense environment at high-z. Thus, we need to infer the distribution, the dynamics, and the ionisation state of LAB 1’s Lyα emitting gas. Methods. LAB 1 was observed for 17.2 h with the VLT/MUSE integral-field spectrograph. We produced optimally extracted narrow band images, in Lyαλ1216, He IIλ1640, and we tried to detect C IVλ1549 emission. By utilising a moment-based analysis, we mapped the kinematics and the line profile characteristics of the blob. We also linked the inferences from the line profile analysis to previous results from imaging polarimetry. Results. We map Lyα emission from the blob down to surface-brightness limits of ≈6 × 10−19 erg s−1 cm−2 arcsec−2. At this depth, we reveal a bridge between LAB 1 and its northern neighbour LAB 8, as well as a shell-like filament towards the south of LAB 1. The complexity and morphology of the Lyα profile vary strongly throughout the blob. Despite the complexity, we find a coherent large-scale east-west velocity gradient of ∼1000 km s−1 that is aligned perpendicular to the major axis of the blob. Moreover, we observe a negative correlation of Lyα polarisation fraction with Lyα line width and a positive correlation with absolute line-of-sight velocity. Finally, we reveal He II emission in three distinct regions within the blob, however, we can only provide upper limits for C IV. Conclusions. Various gas excitation mechanisms are at play in LAB 1: ionising radiation and feedback effects dominate near the embedded galaxies, while Lyα scattering contributes at larger distances. However, He II/Lyα ratios combined with upper limits on C IV/Lyα are not able to discriminate between active galactic nucleus ionisation and feedback- driven shocks. The alignment of the angular momentum vector parallel to the morphological principal axis appears to be at odds with the predicted norm for high-mass halos, but this most likely reflects that LAB 1 resides at a node of multiple intersecting filaments of the cosmic web. LAB 1 can thus be thought of as a progenitor of a present-day massive elliptical within a galaxy cluster.

The use of Kepler solver in numerical integrations of quasi-Keplerian orbits

ABSTRACT A Kepler solver is an analytical method used to solve a two-body problem. In this paper, we propose a new correction method by slightly modifying the Kepler solver. The only change to the analytical solutions is that the obtainment of the eccentric anomaly relies on the true anomaly that is associated with a unit radial vector calculated by an integrator. This scheme rigorously conserves all integrals and orbital elements except the mean longitude. However, the Kepler energy, angular momentum vector, and Laplace–Runge–Lenz vector for perturbed Kepler problems are slowly varying quantities. However, their integral invariant relations give the quantities high-precision values that directly govern five slowly varying orbital elements. These elements combined with the eccentric anomaly determine the desired numerical solutions. The newly proposed method can considerably reduce various errors for a post-Newtonian two-body problem compared with an uncorrected integrator, making it suitable for a dissipative two-body problem. Spurious secular changes of some elements or quasi-integrals in the outer Solar system may be caused by short integration times of the fourth-order Runge–Kutta algorithm. However, they can be eliminated in a long integration time of 108 yr by the proposed method, similar to Wisdom–Holman second-order symplectic integrator. The proposed method has an advantage over the symplectic algorithm in the accuracy but gives a larger slope to the phase error growth.