The method proposed is a generalization of the method described in [1]. It is represented in terms of the Hamiltonian formalism with all the advantages arising from it. A complete presentation of the method is given in the paper [2].

A first-order linear partial differential equation is presented, giving the non-stationary potential functions U=U (x,y,t) which give rise to a given family of evoling planar orbits f(x,y,t) = c in two-dimensional dynamical system. It is shown, that this equation is applied in celestial mechanics of variable mass.

In this paper we have tested some non-standard numericcal methods to integrate the two-body problem. The integration has been performed with Bettis methods, a symplectic algorithm and special spherically exact scheme. The results have been compared with those obtained with traditional integrators.

This paper is devoted to the study of motion of an artificial satellite relative to its centre of mass near parametric resonance in elliptic orbit. It is well known fact that the satellite, of the form of an ellipsoid with three unequal axes, while moving about the central planet, oscillates about the stable position of equilibrium (the longest axis of the satellite coinciding with the radius vector of its centre of mass). The oscillation of the satellite about this position of equilibrium in the orbital plane of its centre of mass is described by a well known second order nonlinear differential equation with a periodic sine force. Naturally there will be resonance cases (main as well as parametric) for such a systems. In the previous author’s work, it was discovered a series of parametric resonances for the system, which corresponds to n = ½k where k is a non-zero integer and n is a parameter depending on the shape of the satellite. The parametric resonance, for k = 1, has been considered here. The first approximate solution of the equation of motion has been obtained by Eogoliubov-Krilov method with e (the eccentricity of the orbit of the centre of mass of the satellite) as the small parameter. This method enables us to visualise the oscillation of the satellite for the resonance case as well as near the resonance. Three stationary values of the amplitudes and phase of oscillation have been obtained, out of which only one is stable near this particular parametric resonance. At the resonance there appear only one stationary regime of oscillation with a very small amplitude.

The theoretical framework adopted in astrophysics and cosmology, in both modelling and the analysis of the observational data, is often implicitly assumed to be that of structural stability. Here, in view of some of the recent results in dynamical systems theory, it is argued that such a framework cannot be assumed a priori and that the fragility framework may instead turn out to be the appropriate framework for the study of certain phenomena in the astrophysical and the cosmological settings. This is motivated by a number of examples from cosmology and a brief discussion of some of the potential domains of its relevance in astrophysics.

It is well known that the Sun’s radiation and a large number of phenomena occurring on the sun have influence on the Earth’s near environment i.e. Atmosphere, Ionosphere, Magnetosphere, etc. These manifest themselves as day-night, seasons, tides and many changes in the neutral atmosphere; changes in meteorological parameters. These changes are directly or indirectly related to variations in solar parameters, such as solar flares, magnetic storms, variations in sunspot number occurring in solar photosphere. Sunsports are observed, their number counted and their accurate records maintained for long time (many centuaries). The sunspot number seems to follow periodic changes with several periods; mainly 11 years and 23.5 years. Recently it has been shown that the combined tidal force of the inner planets and two largest planets, Jupiter and Saturn, have periodic change of 11 and 23.5 years. It was proposed that this small force may be having a tiny influence on the surface of the Sun and causing some nonlinear effect which results into formation of sunspots and thus causes the variations in the number of sunspots. In the present work it is shown that whenever the combined tidal force on sun increases then sunspot number seems to increase and when force decreases sunspot number decreases. This is shown for Solar Cycle number 21.

The orbital instability and forming of outer part of Asteroidal belt has been studied earlier in the Circular case. Here the same problem is studied in elliptic case.