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In the framework of restricted five bodies problem, the existence and stability of the libration points are explored and analysed numerically, under the effect of non--isotropic mass variation of the fifth body (test particle or infinitesimal body). The evolution of the positions of these points and the possible regions of motion are illustrated, as a function of the perturbation parameter. We perform a systematic investigation in an attempt to understand how the perturbation parameter due to variable mass of the fifth body, affects the positions, movement and stability of the libration points. In addition, we have revealed how the domain of the basins of convergence associated with the libration points are substantially influenced by the perturbation parameter.

Tidal interactions between Planet and its satellites are known to be the main phenomena, which are determining the orbital evolution of the satellites. We suggest in the current research to take into consideration the additional well-known effect of differential rotation which obviously takes place in the gaseous or fluid convection zone of primary giant Planet (indeed, the aforementioned effect exists even not depending on the orbital evolution of satellites around host Planet). Nevertheless, estimations for the contribution of the aforementioned effect of differential rotation in the Uranus system (including all its most massive satellites) let us exclude using such effect from calculations of mutual evolution of the eccentricity e along with the semi-major axis a for all satellites of Uranus (Planet of ice-type). It means that the Uranus can be considered in the analytic exploration of governing equations as to be the appropriate candidate for applying the modern ansatz [Efroimsky, 2015] (tidal dissipation effect depending on the tidal-flexure frequency) in regard to estimations of eccentricity e along with semi-major axis a for satellites of Uranus. We can see from the results of calculations in Section 3 that the combined system of governing equations (in the sense of combined contributions to the tidal dissipation from Uranus + from satellite) yields the really observed magnitudes of decelerations for semi-major axises of all the satellites of Uranus. Meanwhile, internal heat generation effects in the satellites (due to tidal dissipation effect) are much more than those which definitely take place in the Uranus, excepting the case of Ariel., Comment: 14 pages, 2 figures; Keywords: Tidal dissipation, differential rotation of Uranus, the Lense-Thirring effect

We offer an analytical study on the dynamics of a two-body problem perturbed by small post-Newtonian relativistic term. We prove that, while the angular momentum is not conserved, the motion is planar. We also show that the energy is subject to small changes due to the relativistic effect. We also offer a periodic solution to this problem, obtained by a method based of separation of timescales. We demonstrate that our solution is more general than the method developed in the book by Brumberg (1991). The practical applicability of this model may be studies of the long-term evolution of relativistic binaries (neutron stars or black holes).

The aim of the present paper is to study the dynamics of a dumbbell satellite moving in a gravity field generated by an oblate body considering the effect of the zonal harmonic parameter. We prove that the pass trajectory of the mass center of the system is periodic and different from the classical one when the effect of the zonal harmonic parameter is non zero. Moreover, we complete the classical theory show- ing that the equations of motion in the satellite approximation can be reduced to Beletsky's equation when the zonal harmonic parameter is zero. The main tool for proving these results is the Lindstedt{Poincare's technique.

The aim of the present work is to reduce the secular solution around the triangular equilibrium points to periodic solution in the frame work of the generalized restricted thee-body problem. This model is generalized in sense that both the primaries are oblate and radiating as well as the gravitational potential from a belt. We show that the linearized equation of motion of the infinitesimal body around the triangular equilibrium points has a secular solution when the value of mass ratio equals the critical mass value. Moreover, we reduce this solution to periodic solution, as well as some numerical and graphical investigations for the effects of the perturbed forces are introduced. This model can be used to examine the existence of a dust particle near the triangular points of an oblate and radiating binary stars system surrounded by a belt.

It is known that Ermakov-Pinney equation is a nonlinear equation with wide applications in dynamics, physics, cosmology (e.g., Ermakov equation can be connected to Bose-Einstein Condensate cosmology which unifies the dark energy and the dark matter). In this analytical study, we have presented a new type of solving procedure to obtain analytical solution of Ermakov-Pinney equation, specifically for the case of rotating early Universe with vortex. The particular case of solution of the aforementioned equation is presented also (such the solution of special kind is important for cosmological applications) which corresponds to the class of solutions with symmetry reduction., Comment: 11 pages; Keywords: Ermakov-Pinney equation, Riccati equation, Bose-Einstein Condensate cosmology, dark energy, dark matter, symmetry reduction