Quaternion Regularization of Singularities of Astrodynamic Models Generated by Gravitational Forces (Review)

Abstract

The article presents an analytical review of works devoted to the quaternion regularization of the singularities of differential equations of the perturbed three-body problem generated by gravitational forces, using the four-dimensional Kustaanheimo–Stiefel variables. Most of these works have been published in leading foreign publications. We consider a new method of regularization of these equations proposed by us, based on the use of two-dimensional ideal rectangular Hansen coordinates, two-dimensional Levi-Civita variables, and four-dimensional Euler (Rodrigues–Hamilton) parameters. Previously, it was believed that it was impossible to generalize the famous Levi-Civita regularization of the equations of plane motion to the equations of spatial motion. The regularization proposed by us refutes this point of view and is based on writing the differential equations of the perturbed spatial problem of two bodies in an ideal coordinate system using two-dimensional Levi-Civita variables to describe the motion in this coordinate system (in this coordinate system, the equations of spatial motion take the form of equations of plane motion) and based on the use of the quaternion differential equation of the inertial orientation of the ideal coordinate system in the Euler parameters, which are the osculating elements of the orbit, as well as on the use of Keplerian energy and real time as additional variables, and on the use of the new independent Sundmann variable. Reduced regular equations, in which Levi-Civita variables and Euler parameters are used together, have not only the well-known advantages of equations in Kustaanheimo–Stiefel variables (regularity, linearity in new time for Keplerian motions, proximity to linear equations for perturbed motions), but also have their own additional advantages: 1) two-dimensionality, and not four-dimensionality, as in the case of Kustaanheimo-Stiefel, a single-frequency harmonic oscillator describing in new time in Levi-Civita variables the unperturbed elliptic Keplerian motion of the studied (second) body, 2) slow change in the new time of the Euler parameters, which describe the change in the inertial orientation of the ideal coordinate system, for perturbed motion, which is convenient when using the methods of nonlinear mechanics. This work complements our review paper [1].