Quaternion Regularization of Differential Equations of Perturbed Central Motion and Regular Models of Orbital (Trajectory) Motion: Review and Analysis of Models, Their Applications

The review article briefly outlines our proposed general quaternion theory of regularizing and stabilizing transformations of Newtonian differential equations of perturbed motion of a material point in a central force field, the potential of which is assumed to be an arbitrary differentiable function of the distance from the point to the center of the field. The point is also under the influence of a disturbing potential, assumed to be an arbitrary function of time and Cartesian coordinates of the point’s location, and under the influence of a disturbing acceleration, assumed to be an arbitrary function of time, the radius vector and the point’s velocity vector. The conditions for the reducibility of the presented quaternion equations of perturbed central motion to an oscillatory form are considered using three regularizing functions containing the distance to the center of the field. Various differential quaternion equations of perturbed central motion in oscillatory and normal forms, constructed using this theory, are presented, including regular equations that use four-dimensional Euler (Rodrigue–Hamilton) parameters or four-dimensional Kustaanheim–Stiefel variables or their modifications, proposed by us. Regular quaternion equations of spatial unperturbed central motion of a material point, connections of the four-dimensional variables used with orbital elements, and a uniformized solution to the spatial problem of unperturbed central motion are considered. As an application, regularized differential quaternion equations of motion of an artificial satellite in the Earth’s gravitational field are presented in four-dimensional Kustaanheimo-Stiefel variables, as well as in our modified four-dimensional variables and in Euler parameters. An analysis of the stated regular quaternion equations of perturbed central motion is presented, showing that the quaternion regularization method, based on the use of Euler parameters or Kustaanheim–Stiefel variables or their modifications, is unique in joint regularization, linearization and increase in dimension for three-dimensional Keplerian systems and central movement.

Presented regularized (with respect to the Newtonian force of attraction) differential quaternion equations of motion of an artificial satellite in the gravitational field of the Earth in our modified four-dimensional variables have the advantages indicated in the article over quaternion equations in the Kustaanheimo–Stiefel variables. In the presented differential quaternion equations of satellite motion, constructed using four-dimensional Euler parameters, the terms of the equations containing negative powers of the distance to the center of the Earth of the fourth order, inclusive, are regularized. In all these regularized equations, the description of the Earth’s gravitational field takes into account not only the central (Newtonian), but also the zonal, tesseral and sectorial harmonics of the potential of the Earth’s gravitational field (the nonsphericity of the Earth is taken into account).

Keywords: differential equations of the perturbed central motion of a material point, quaternion theory of regularizing and stabilizing transformations, conditions for the reducibility of quaternion equations to oscillatory form, regularizing functions, quaternion equations in oscillatory and normal forms, Euler (Rodrigue–Hamilton) parameters, Kustaanheimo–Stiefel variables, modified four-dimensional variables, orbital elements, uniform solution, regularized quaternion equations of satellite motion; central (Newtonian), zonal, tesseral and sectorial harmonics of the Earth’s gravitational field potential