 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
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| Yu.N. Chelnokov, "Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: II," Mech. Solids. 53 (6), 633-650 (2018) |
| Year |
2018 |
Volume |
53 |
Number |
6 |
Pages |
633-650 |
| DOI |
10.3103/S0025654418060055 |
| Title |
Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: II |
| Author(s) |
Yu.N. Chelnokov (Institute of Precision Mechanics and Control Problems of the Russian Academy of Sciences, ul. Rabochaya 24, Saratov, 410028 Russia, ChelnokovYuN@gmail.com) |
| Abstract |
A quaternion method for the regularization of differential equations of the perturbed spatial restricted three-body problem is developed. It is closely related, from the methodological point of view, to the quaternion method for the regularization of the differential equations of the perturbed spatial three-body problem in Kustaanheimo-Stiefel variables that was earlier proposed by the author of this article.
Various local and global regular quaternion differential equations of the perturbed spatial restricted three-body problem (both circular and non-circular problem) i.e. equations that are regular in the vicinity of the first or second body of finite mass and equations that are regular at the same time both in the neighborhood of the first and second body of finite mass are obtained. The equations are systems of nonlinear nonstationary differential equations of the tenth or eleventh or nineteenth order with respect to the Kustaanheimo-Stiefel variables, their first derivatives, Kepler or total energies, or variables that are Jacobi integration constants in the case of the unperturbed spatial circular restricted three-body problem, as well as with respect to time and auxiliary time variable. The equations obtained allow one to construct different regular algorithms for integrating the differential equations of the perturbed spatial restricted three-body problem.
This study is an extension of [1, 2]. |
| Keywords |
non-circular and circular three-body problems, differential equations of motion, regularization, quaternion, Kustaanheimo-Stiefel variables, energy, Jacobi integral, time transformation |
| References |
| 1. | Yu.N. Chelnokov,
"Quaternion Regularization for the Equations of the Two-Body and Restricted Three-Body Problems,"
in Proc. of 9-th All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics
(Izd-vo Kazan Federal. University Kazan', 2015), pp. 4051-4053 [in Russian]. |
| 2. | Yu.N. Chelnokov,
"Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I,"
Izv. Akad. Nauk. Mekh. Tv. Tela,
No. 6, 24-54 (2017)
[Mech. Sol. (Engl. Transl.)
52 (6), 613-639 (2017)]. |
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(Nauka, Moscow, 1983) [in Russian]. |
| 5. | Yu.N. Chelnokov,
"To Regularization of Equations of Spatial Two-Body Problem,"
Izv. Akad. Nauk SSSR. Mekh. Tv. Tela,
No. 6, 12-21 (1981) [Mech. Sol. (Engl. Transl.)]. |
| 6. | Yu.N. Chelnokov,
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Izv. Akad. Nauk SSSR. Mekh. Tv. Tela,
No. 1, 151-158 (1984) [Mech. Sol. (Engl. Transl.)]. |
| 7. | Yu.N. Chelnokov,
"Quaternion Regularization and Stabilization of Perturbed Central Motion. Pt. 1,"
Izv. Ross. Akad. Nauk. Mekh. Tv. Tela,
No. 1, 20-30 (1993) [Mech. Sol. (Engl. Transl.)]. |
| 8. | Yu.N. Chelnokov,
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Izv. Ross. Akad. Nauk. Mekh. Tv. Tela,
No. 2, 3-15 (1993) [Mech. Sol. (Engl. Transl.)] |
| 9. | Yu.N. Chelnokov,
"Analysis of Optimal Motion Control for a Material Point in a Central Field with Application of Quaternions,"
Izv. Akad. Nauk. Teor. Sist. Upr.
No. 5, 18-44 (2007)
[J. Comp. Sys. Sci. Int. (Engl. Transl.)
46 (5) 688-713 (2007)]. |
| 10. | Yu.N. Chelnokov,
Quaternion Models and Methods of Dynamics, Navigation, and Control of Motion
(Fizmatlit, Moscow, 2011) [in Russian]. |
| 11. | Yu.N. Chelnokov,
"Quaternion Regularization in Celestial Mechanics and Astrodynamics and Trajectory Motion Control. I,"
Kosmich. Issled. 51 (5), 389-401 (2013)
[Cosmic Res. (Engl. Transl.) 51 (5), 350-361 (2013)]. |
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Application of Quaternions in Problems of Attitude Control of a Rigid Body
(Nauka, Moscow, 1973) [in Russian]. |
| 14. | A.Yu. Ishlinskii,
Orientation, Gyroscopes, and Inertial Navigation
(Nauka, Moscow, 1976) [in Russian]. |
| 15. | Yu.N. Chelnokov,
Quaternion and Biquaternion Models and Methods of Mechanics of Solids and Their Applications
(Fizmatlit, Moscow, 2006) [in Russian] |
| 16. | V.Ph. Zhuravlev,
Foundations of Theoretical Mechanics
(Fizmatlit, Moscow, 2008) [in Russian]. |
| 17. | P. Kustaanheimo,
"Spinor Regularization of the Kepler Motion,"
Ann. Univ. Turk.
Ser. A. 73, 3-7 (1964). |
| 18. | P. Kustaanheimo and E. Stiefel,
"Perturbation Theory of Kepler Motion Based on Spinor Regularization,"
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| 19. | E. Stiefel and G. Scheifele,
Linear and Regular Celestial Mechanics
(Springer, Berlin, 1971; Nauka, Moscow, 1975). |
| 20. | Yu.N. Chelnokov,
"Application of Quaternions in Theory of Orbital Motion of Artificial Satellite. II,"
Kosm. Issled.
31 (3), 3-15 (1993)
[Cosmic Res. (Engl. Transl.)
31 (6), 409-418 (1992)]. |
| 21. | R. Roman and I. Szucs-Csillik,
"Generalization of Levi-Civita Regularization in the Restricted Three-Body Problem,"
Astrophys. Space Sci.
349, 117-123 (2014). |
| 22. | S.J. Aarseth and K. Zare
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Cel. Mech.
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Gravitational N-Body Simulations
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Contemporary Numerical Methods in Problems of Celestial Mechanics
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| 25. | T.V. Bordovitsyna, and V.A. Avdyushev,
Theory of Motion of Artificial Satellites of the Earth. Analytical and Numerical Methods
(Izd-vo Tomsk. Univ., Tomsk, 2007) [in Russian]. |
|
| Received |
03 February 2017 |
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