Research into the Dynamics of a System of Two Connected Bodies Moving in the Plane of a Circular Orbit by Applying Computer Algebra Methods

Abstract

Computer algebra methods are used to determine the equilibrium orientations of a system of two bodies connected by a spherical hinge that moves in a central Newtonian force field on a circular orbit under the action of gravitational torque. Primary attention is given to the study of equilibrium orientations of the two-body system in the plane of the circular orbit. By applying symbolic differentiation, differential equations of motion are derived in the form of Lagrange equations of the second kind. A method is proposed for transforming the system of trigonometric equations determining the equilibria into a system of algebraic equations, which in turn are reduced by calculating the resultant to a single algebraic equation of degree 12 in one unknown. The roots of the resulting algebraic equation determine the equilibrium orientations of the two-body system in the circular orbit plane. By applying symbolic factorization, the algebraic equation is decomposed into three polynomial factors, each specifying a certain class of equilibrium configurations. The domains with an identical number of equilibrium positions are classified using algebraic methods for constructing a discriminant hypersurface. The equations for the discriminant hypersurface determining the boundaries of domains with an identical number of equilibrium positions in the parameter space of the problem are obtained via symbolic computations of the determinant of the resultant matrix. By numerical analysis of the real roots of the resulting algebraic equations, the number of equilibrium positions of the two-body system is determined depending on the parameters.