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In classical celestial mechanics, the theory of perturbation based on Keplerian motion is well developed. Various kinds of perturbed motion equations are described by canonical equations, Lagrange equations and Newton's equations. The masses of bodies are assumed constant and the bodies are point or spherical. Classical perturbation theory is a basic tool in the study of many astronomical problems. Observational astronomy shows that the masses of real cosmic bodies change over time. Real celestial bodies are neither spherical nor solid. Celestial bodies are non-stationary and their masses, sizes, shapes and structures change over time as they evolve. There are now a number of studies with a community bibliography of papers on the celestial mechanics of bodies of variable mass. However, the perturbation theory for unsteady space systems is not sufficiently developed. Therefore, we considered a gravitational system consisting of three spherical celestial bodies, with spherical density distributions, with variable masses in the restricted formulation. We have considered the general case where the masses of the bodies change non-isotropically at different rates, in the presence of reactive forces. We will assume that a body with infinitesimal mass has no effect on the motions of the two main bodies. The problem was investigated by methods of perturbation theory based on aperiodic motion along the quasi-conic section developed for nonstationary gravitating systems. Evolution motion equations of a less massive body and a small body in oscillating elements, in the relative coordinate system, have been obtained.

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Numerical studies of astrophysical objects are a relatively new direction in Fesenkov Astrophysical Institute (FAI) and is mainly represented by the Laboratory of Cosmology, Stellar Dynamics and Computational Astrophysics. The lab seeks to understand the evolution of gravitating systems at various scales – from star clusters to galaxies to large-scale structure of the universe as a whole, and tackles these problems both through analytical methods and through numerical simulations. The particular focus is on numerical simulations of star clusters, especially those found in active galactic nuclei – this is a topic of oldestablished collaboration with colleagues from Astronomisches Rechen-Institut (Heidelberg) and National Astronomical Observatories of China (Beijing). The prominent example is STARDISK project dedicated to the numerical research of active galactic nuclei as multicomponent systems composed of compact stellar cluster, gaseous accretion disk and a supermassive black hole. It is demonstrated that an accretion disk can noticeably decelerate stars and thus enhance the accretion rate onto the black hole. In 2013 FAI hosted the MODEST-13 International Workshop dedicated to modeling of star clusters. Recently a new project has been approved aimed at construction of triaxial equilibrium N-body systems that can be of great help in various numerical experiments with disk galaxies. There are also long standing plans to perform cosmological simulations of large scale structures to test a new approach to dark matter and energy actively developed at FAI. For numerical calculations, FAI has a small, but growing computer cluster consisting of several high-performance computing servers equipped with computational GPU cards.