Periodic motions of a close to dynamically symmetrical satellite in the neighbourhood of a conical precession

Abstract

The motions of a close to dynamically symmetrical satellite in a circular orbit, that is, of a rigid body in a central Newtonian gravitational field, are considered. The periodic motions, generated from the conical precession of a dynamically symmetrical satellite, are constructed in the unperturbed problem. A rigorous, non-linear analysis of the stability of these motions is carried out. In the unperturbed problem, one of the coordinates, the angle of natural rotation of the satellite, is cyclic and the system of differential equations describing the motions of the perturbed problem is close to the system with the cyclic coordinate. The resonant case, when the ratio of one of the frequencies of small oscillations of the reduced system in the neighbourhood of a stable equilibrium to the frequency of the change in the cyclic coordinate is close to an integer and the case when there is no resonance are investigated. Previously obtained [1] results of an investigation of the periodic motions of autonomous Hamiltonian systems with two degrees of freedom are extended to the case of a system with three degrees of freedom being considered here, when the above-mentioned resonance is present. When there is no such resonance, the cases of parametric resonance, of third- and fourth-order resonance and, also, the general non-resonant case are distinguished. Results for the stability of non-autonomous Hamiltonian systems with two degrees of freedom in the case of resonances [2] and, also, the results of KAM-theory (in the general non-resonant case) [3] are used.