Quaternionic Regular Equations of the Two-Body Problem and the Problem of the Motion of a Satellite in the Gravitational Field of the Earth in Kustaanheim-Stiefel Variables and Modified Four-Dimensional Variables: Dynamics of Relative Motion

The article develops the quaternion regularization of differential equations (DE) of the relative perturbed motion of the body under study, which we previously proposed within the framework of the perturbed spatial problem of two bodies: the equations of motion of the center of mass of this body in a coordinate system rotating in an inertial coordinate system according to an arbitrarily given law, and also develops a quaternion DE regularization of the motion of the body under study relative to the coordinate system associated with the Earth. New quaternion DEs of the perturbed motion of an artificial Earth satellite relative to the coordinate system associated with the Earth are proposed. These equations have (in modern times) the form of DE of the relative motion of the perturbed four-dimensional oscillator in the Kustaanheimo-Stiefel variables or in our proposed modified four-dimensional variables, supplemented by DE for the energy of the satellite motion and time. These equations for the perturbed relative motion of the satellite take into account the zonal, tesseral and sectorial harmonics of the Earth's gravitational field. The proposed equations, in contrast to classical equations, are regular (do not contain special points such as singularity (division by zero)) for the relative motion of a satellite in the Newtonian gravitational field of the Earth. The equations are convenient for applying methods of nonlinear mechanics and high-precision numerical calculations when studying the orbital motion of a satellite relative to the Earth and predicting its motion.