Abstract
This paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.
Highlights
We present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame
Stiefel and Scheifele[37], Bordovitsyna[38], Bordovitsyna and Avdyushev[39], Fukushima[40,41], Pelaez et al.[42], Bau et al.[43], Amato et al.[44], and Bau and Roa[45] demonstrated the results of comparing the numerical solutions to the equations of orbital motion of celestial and cosmic bodies in the KS variables, Euler parameters, and other variables. These results prove the efficiency of the KS variables and Euler parameters in celestial mechanics and astrodynamics
Note that our work described regular quaternion differential equations of the perturbed spatial two-body problem, which are free from singularities induced by gravitational forces
Summary
Celestial mechanics and astrodynamics are based on Newtonian differential equations of the perturbed spatial two-body problem and the perturbed spatial restricted three-body problem. This relation, according to Stiefel and Scheifele[37], is essential to their formulation of regular celestial mechanics These interpretations were offered by Chelnokov[20,21], who showed that the regularizing KS transformation of the coordinates consists of transitioning from the Cartesian coordinates of the second body to the normed Euler parameters (components of the Hamilton rotation quaternion). The conclusion sums up our research in the field of quaternion regularization of the perturbed spatial two-body problem equations
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