On the Dynamics of a Gravitational Dipole

A. P. Markeev

Russian journal of nonlinear dynamics, Tome 17 (2021) no. 3, pp. 247-261

Résumé

An orbital gravitational dipole is a rectilinear inextensible rod of negligibly small mass which moves in a Newtonian gravitational field and to whose ends two point loads are fastened. The gravitational dipole is mainly designed to produce artificial gravity in a neighborhood of one of the loads. In the nominal operational mode on a circular orbit the gravitational dipole is located along the radius vector of its center of mass relative to the Newtonian center of attraction. The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a neighborhood of its nominal mode. The orbit of the center of mass is assumed to be circular or elliptic with small eccentricity. Consideration is given both to planar and arbitrary spatial deviations of the gravitational dipole from its position corresponding to the nominal mode. The analysis is based on the classical Lyapunov and Poincaré methods and the methods of Kolmogorov – Arnold – Moser (KAM) theory. The necessary calculations are performed using computer algorithms. An analytic representation is given for conditionally periodic oscillations. Special attention is paid to the problem of the existence of periodic motions of the gravitational dipole and their Lyapunov stability, formal stability (stability in an arbitrarily high, but finite, nonlinear approximation) and stability for most (in the sense of Lebesgue measure) initial conditions.

Détail

Keywords: nonlinear oscillations, resonance, stability, canonical transformations.