Application of Computer Algebra Methods for Investigation of the Stationary Motions of the System of Two Connected Bodies Moving along a Circular Orbit

Abstract

Computer algebra and numeric methods are used to investigate properties of a nonlinear algebraic system that determines the equilibrium orientations for a system of two bodies connected by a spherical hinge that moves along a circular orbit under the action of gravitational torque. The main attention is paid to study the conditions of existence of the equilibrium orientations for the system of two bodies for special cases, when one of the principal axes of inertia both the first and second body coincides with the normal to the orbital plane, with radius vector or tangent to the orbit. To determine the equilibrium orientations for the system of two bodies, the system of 12 stationary algebraic equations is decomposed into 9 subsystems. The computer algebra method based on the algorithm for the construction of a Gröbner basis applied to solve the stationary motion system of algebraic equations. Depending on the parameters of the problem, the number of equilibria is found by numerical analysis of the real roots of the algebraic equations from the Gröbner basis constructed.