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.

RTA Mathematic Simulation Principles at the Autotechnical Expertise

The features of carrying out an autotechnical expertise (ATE) are considered in case the vehicles (V) participating in the road transport accident (RTA) don’t leave skid imprints. The examples of momentum and energy conservation law application are given at developing the road accident mathematical model. Special attention is paid to the determination methods of vehicle (V) velocity, travel directions in various RTA diagrams and archeology of deformation. For this purpose it is offered to draw a momentum vector diagram. It is reasonable that for the calculation of V deformation at RTA it is necessary to determine step by step the strain-stress state in a contact area on the basis of the theories of elasticity, plasticity, solid friction and finite-element methods. The technique of constructing an RTA mathematical model is developed. It is recommended to use at ATE of RTAs at the runs-over into the fixed obstacle (a stationary V) and collisions.

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.

Spacetime Positive Mass Theorems for Initial Data Sets with Non-Compact Boundary

In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is light-like. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.

Retinal optic flow during natural locomotion

We examine structure of visual motion on the retina during natural locomotion in real world environments. We demonstrate that eye-movement-free/head-centered optic flow is highly unstable due to the complex, phasic head movements that occur throughout the gait cycle. In contrast, VOR-mediated retinal optic flow has stable, reliable features that may be valuable for the control of locomotion. In particular, the sign and magnitude of the curl of retinal flow at the fovea specifies the degree to which a walker will pass to the left or right of their fixation point. In addition, the peak in the divergence of the retinal flow field specifies the walker's overground velocity/momentum vector in retinotopic coordinates. If we assume a walker can reliably determine the body-relative position of their fixation, this retinotopic cue for their body's momentum could be an essential aspect of the visual control locomotion over complex terrain.Clickable Video Links (Click here for a playlist of all videos)https://www.youtube.com/playlist?list=PLWxH2Ov17q5HRHVngfuMgMZn8qfOivMafVideo 1. Gaze Laser Skeleton – Video (Full Speed) – Free Walking – Raw (See Figure 1)Video 2. Gaze Laser Skeleton – Video (1/4 Speed) – Free Walking – Optic Flow VectorsVideo 3. Gaze Laser Skeleton – Video (1/4 Speed) – Free Walking – Optic Flow StreamlinesVideo 4. Sim. Eye Trajectory – Sim. Retinal Flow – Fixation Aligned with PathVideo 5. Sim. Eye Trajectory – Sim. Retinal Flow – Fixation to Left of PathVideo 6. Sim. Eye Trajectory – Sim. Retinal Flow – Fixation to Right of PathVideo 7. Sim. Eye Trajectory – Sim. Retinal Flow – Vertical Sin WaveVideo 8. Sim. Eye Trajectory – Sim. Retinal Flow – Horizontal Sin WaveVideo 9. Sim. Eye Trajectory – Sim. Retinal Flow – CorckscrewVideo 10. Gaze Laser Skeleton – Sim. Retinal Flow – Ground LookingVideo 11. Gaze Laser Skeleton – Video (Full Speed) – Rocky Terrain – RawVideo 12. Gaze Laser Skeleton – Video (1/4 Speed) – Rocky Terrain – Optic Flow StreamlinesVideo 13. Gaze Laser Skeleton – Sim. Retinal Flow – Rocky TerrainVideo 14. Quadcopter Gimbal – Video (Full Speed) – Optic Flow Streamlines

Track reconstruction and momentum measurement

The reconstruction of a charged particle’s trajectory in a magnetic field allows us to determine the momentum vector, that is both the particle’s momentum and its direction. This chapter begins with the description of typical magnetic field configurations and the parametrisation of particle trajectories in homogeneous magnetic fields. For different detector configurations a detailed discussion of the reconstruction of particle trajectories from the measured points. Special attention is given to the achievable resolutions of position, direction, momentum and impact parameter, including newly derived formulas for the effects of multiple scattering.

Rocking shadows in broken circumbinary discs

ABSTRACT We use three-dimensional simulations with coupled hydrodynamics and Monte Carlo radiative transfer to show that shadows cast by the inner disc in broken circumbinary discs move within a confined range of position angles on the outer disc. Over time, shadows appear to rock back and forth in azimuth as the inner disc precesses. The effect occurs because the inner disc precesses around a vector that is not the angular momentum vector of the outer disc. We relate our findings to recent observations of shadows in discs.