26 ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2020, Vol. 60, No. 1, pp. 26–35. © Pleiades Publishing, Ltd., 2020. Russian Text © The Author(s), 2020, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2020, Vol. 60, No. 1, pp. 29–38. A Study of Secular Perturbations of Translational-Rotational Motion in a Nonstationary Two-Body Problem Using Computer Algebra S. B. Bizhanova a , M. Zh. Minglibayev a,b , and A. N. Prokopenya c, * a Al-Farabi Kazakh National University, Almaty, 050040 Kazakhstan b Fesenkov Astrophysical Institute, Almaty, 050020 Kazakhstan c Warsaw University of Life Sciences, Warsaw, 02-776 Poland *e-mail: alexander_prokopenya@sggw.pl Received July 29, 2019; revised July 29, 2019; accepted September 18, 2019 Abstract—A nonstationary two-body problem is considered such that one of the bodies has a spheri- cally symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The body masses vary iso- tropically at different rates. Equations of motion of the satellite in a relative system of coordinates are derived. The problem is studied by the methods of perturbation theory. Equations of secular perturba- tions of the translational-rotational motion of the satellite in analogues of Delaunay–Andoyer oscu- lating elements are deduced. All necessary symbolic computations are performed using the Wolfram Mathematica computer algebra system. Keywords: variable mass, secular perturbations, axisymmetric body, translational-rotational motion DOI: 10.1134/S0965542520010054 1. INTRODUCTION Present-day observational data in astronomy show that real cosmic systems are nonstationary: their masses, sizes, shapes, and a number of other physical characteristic vary with time in the process of evo- lution (see [1–6]). In this regard, creating mathematical models of motion of nonstationary celestial bod- ies becomes relevant. The purpose of this study is to derive differential equations of the translational-rotational motion of a nonstationary axisymmetric body of variable mass and sizes and of variable oblateness in a nonstationary central gravitational field. Solving this problem involves rather cumbersome symbolic computations, which are best to perform using computer algebra systems (see [7, 8]). In this work, all necessary symbolic computations are carried out using the Wolfram Mathematica system [9, 10]. 2. PROBLEM STATEMENT AND EQUATIONS OF MOTION IN A RELATIVE SYSTEM OF COORDINATES Assume that the first central body T 1 with variable mass m 1 = m 1 (t) is a ball with a spherically symmetric density distribution. Assume that the second body T 2 with mass m 2 = m 2 (t) is a satellite of T 1 and has an axisymmetric dynamical structure and shape and that its second-order moments of inertia are given func- tions of time. Such a satellite is characterized by variable oblateness, and its principal central moments of inertia A, B, and C satisfy the relations (1) Assume that the laws of body mass variations are known and that the masses vary at different rates, that is, (2) - = ≠ ≠ () () () () ( ), const. () Ct At At Bt Ct Ct ≠ ɺ ɺ 1 2 1 2 . m m m m