Abstract:
The relative motion between multiple satellites is a developed technique with many applications. Formation-flying missions use the relative motion dynamics in their design. In this work, the motion in invariant relative orbits is considered under the effects of second-order zonal harmonics in an equatorial orbit. The Hamiltonian framework is used to formulate the problem. All the possible conditions of the invariant relative motion are obtained with different inclinations of the follower satellite orbits. These second-order conditions warrantee the drift rates keeping two, or more, neighboring orbits from drifting apart. The conditions have been modeled. All the possibilities of choosing mean elements of the leader satellite orbit and differences in momenta between leader and follower satellites’ orbits are presented.

Abstract:
The paper draws attention to the importance of the notion “luminiferous ether” in physics. There is a proposed method to register its flows generated by natural cosmic movements or created artificially. The work presents the results of ether wind searching with a prototype of the proposed installation located at the altitude of <30 m above sea level. Ether flows with speeds > 20 km/s are not found, which is consistent with the results of previous experiments.

Abstract:
In this paper, a model of a leader-follower spacecraft formation in six degrees of freedom is derived and presented. The nonlinear model describes the relative translational and rotationalmotion of the spacecraft, and extends previous work by providing a more complete factorization, together with detailed information about the matrices in the model. The resulting model shows many similarities with models for systems such as robot manipulators and marine vehicles. In addition, mathematical models of orbital perturbations due to gravitational variations, atmospheric drag, solar radiation and third-body effects are presented for completeness. Results from simulations are presented to visualize the properties of the model and to show the impact of the different orbital perturbations on the flight path.

Abstract:
Precise vertical total electron contents (VTEC) and its time variation have been obtained by using GPS dual-frequency observations collected by the continuously operating GPS tracking stations distributed over China. Using VTEC data, the wave-motion appearing in the ionosphere on November 3, 2003 is monitored and analyzed when a small solar flare happened. Detailed discussion with the VTEC and its change rate series, which are derived from the observations (data) from PRN23 satellite, indicates that the wave-motion mainly contains two dominant frequencies and propagates almost along the meridian line toward south. Additionally, the fluctuation of the mean VTEC has been calculated in a regional Single Layer Model (SLM) ionospheric shell in the range of N28.0 ° – 34.0 ° and E118.0 ° – 123.0 °. The spectral analysis and the multi-resolution analysis of mean VTEC time series show that the periods of these two components of the wave-motion at middle altitude are around 60 and 25 min separately, and the amplitudes can be up to 1.0–2.0 TECU and 0.4–0.7 TECU respectively. Meanwhile, the relative motion between the wave-motion and the Ionosphere Pierce Points (IPPs), which are defined as the cross points between the line-of-sight of the GPS signals and the SLM thin shell, reveals that the traveling speeds of the two components are about 120–150 m/s and 30–40 m/s respectively.

Abstract:
The human visual perception of motion in depth is based on thespecific sensitivities to two-dimensional relative motion on the fron-tal plane. Several psychophysical experiments were carried out todisclose the mathematical and physiological aspects of these sensitivi-ties. It was concluded that the human visual system contains neu-rons tuned to relative motion on the frontal plane, which formedthe mechanisms of recovering motion in depth. The psychophysicaland physiological evidences were shown for the monocular and bino-cular perceptions of motion in depth. The difference between visualperceptions of motion in depth of humans and those of insects werealso discussed.

Abstract:
This paper discusses in detail the conformal invariance by infinitesimal transformations of a dynamical system of relative motion. The necessary and sufficient conditions of conformal invariance and Lie symmetry are given simultaneously by the action of infinitesimal transformations. Then it obtains the conserved quantities of conformal invariance by the infinitesimal transformations. Finally an example is given to illustrate the application of the results.

Abstract:
Lie symmetry and the generalized Hojman conserved quantity of Nielsen equations for a variable mass holonomic system of relative motion are studied. The determining equation of Lie symmetry of Nielsen equations for a variable mass holonomic system of relative motion under the infinitesimal transformations of groups is given. The expression of generalized Hojman conserved quantity deduced directly from Lie symmetry for a variable mass holonomic system of relative motion is obtained. An example is given to illustrate the application of the results.

Abstract:
The Lie symmetry and Hojman conserved quantity of Nielsen equations in a dynamical system of relative motion with nonholonomic constraint of the Chetaev type are studied. The differential equations of motion of the Nielsen equation for the system, the definition and the criterion of Lie symmetry, and the expression of the Hojman conserved quantity deduced directly from the Lie symmetry for the system are obtained. An example is given to illustrate the application of the results.

Abstract:
Special Lie symmetry and the Hojman conserved quantity for Appell equations in a dynamical system of relative motion are investigated. The definition and the criterion of the special Lie symmetry of Appell equations in a dynamical system of relative motion under infinitesimal group transformation are presented. The expression of the equation for the special Lie symmetry of Appell equations and the Hojman conserved quantity, deduced directly from the special Lie symmetry in a dynamical system of relative motion, are obtained. An example is given to illustrate the application of the results.

Abstract:
This paper focuses on studying the Poisson theory and the integration method of dynamics of relative motion. Equations of a dynamical system of relative motion in phase space are given. Poisson theory of the system is established. The Jacobi last multiplier of the system is defined, and the relation between the Jacobi last multiplier and the first integrals of the system is studied. Our research shows that for a dynamical system of relative motion, whose configuration is determined by n generalized coordinates, the solution of the system can be found by using the Jacobi last multiplier if (2n-1) first integrals of the system are known. At the end of the paper, an example is given to illustrate the application of the results.