ON THE BELETSKY EQUATION

This paper was presented at the Beletsky Session of the 4th IAA Conference on University Satellite Missions and CubeSat Workshop held in Rome and dedicated to the memory of the great Russian mathematician and pioneer scientist of Astrodynamics. The paper is concentrated to the so called Beletsky equation, concerning the attitude motion of a satellite under gravity gradient torque. In fact Professor Beletsky dealt with many aspects of astrodynamics giving deep and useful contributions and establishing some theoretical basis for the development of the space activity in USSR, then it looks like disappointing that his name is so strictly linked to a simple equation such as(1 + e cos theta)theta '' - 2e sin theta ' + alpha sin theta = 4e sin thetadescribing the planar attitude motion of a satellite in elliptic orbit: The reason of the enormous success of the Beletsky equation is just in its simplicity and in the interesting characteristics of its phase space, where regular and periodic solutions are merged together with unstable and chaotic solutions. This is not unusual for a non linear differential equation, but in the Beletsky equation these characteristics can be understood in deep and the transition to chaos can be checked by various indices, so many researchers were attracted by the results that can be achieved from the abstract point of view of dynamical system theory while obtaining output of concrete interest in space applications. Most of the content presented in this paper is derived from my PhD thesis "Chaos in Astrodymanics" presented at the School of Aerospace Engineering of Rome in 1991 and developed under the supervision of Professor Filippo Graziani.