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Yu.N. Chelnokov, "Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I," Mech. Solids. 52 (6), 613-639 (2017)
Year 2017 Volume 52 Number 6 Pages 613-639
DOI 10.3103/S0025654417060036
Title Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I
Author(s) Yu.N. Chelnokov (Institute for Precision Mechanics and Control Problems of the Russian Academy of Sciences, ul. Rabochaya 24, Saratov, 410028 Russia; Chernyshevskii Saratov State University, ul. Astrakhanskaya 83, Saratov, 410012 Russia, chelnokovyun@gmail.com)
Abstract We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo-Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper.

A survey of papers related to the regularization of the differential equations of the two- and three-body problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues-Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given.

Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo-Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventh-order differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass.

The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classical methods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.
Keywords two- and three-body problems, differential equations of motion, quaternion, Kustaanheimo-Stiefel variables
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Received 30 March 2015
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