Application of the Jacobi integral for the investigation of satellite librations around the centre of mass in a gravitational field

This paper deals with a three-dimensional rotationally and dynamically symmetrical satellite. The centre of mass of the satellite moves in a circular orbit. The existence of two first integrals of motion enables one to transform the system of differential equations to a special form facilitating the choice of the zero-approximation solution. The angles of precession and nutation as well as the amplitude functionk ²(t) are taken as variables of the motion. The first approximation solution is constructed for the case of spatial libration of the satellite axis of dynamical symmetry about the position of stable equilibrium. The series representing the functionk ²(t) is fast convergent due to the fast convergence of the expansions for elliptic functions.