Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

Ying-Jing Qian; Xiao-Dong Yang; Guan-Qiao Zhai; Wei Zhang

doi:10.1007/s10509-017-3115-y

Astrophysics and Space Science

August 2017, 362:136 | Cite as

Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

Authors
Authors and affiliations

Ying-Jing Qian
Xiao-Dong YangEmail author
Guan-Qiao Zhai
Wei Zhang

Original Article

First Online: 24 July 2017

Received: 16 January 2017

Accepted: 09 June 2017

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3 Citations

Abstract

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The $\zeta $-component motion is treated as the dominant motion and the $\xi$ and $\eta $-component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the $\zeta $-position and $\zeta $-velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on $\zeta $-direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.

Keywords

Triangular libration points Vertical periodic orbits Polynomial expansion method

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Notes

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant Nos. 11402007 and 11672007. the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

Appendix

$$\begin{aligned} &{O_{200} = - \frac{21C^{2}}{16} + \frac{21}{8}\mu C^{2} + \frac{3}{8}\sqrt{3} CS + \frac{33S^{2}}{16} - \frac{33}{8}\mu S^{2}, }\\ &{O_{020} = \frac{33C^{2}}{16} - \frac{33}{8}\mu C^{2} - \frac{3}{8}\sqrt{3} CS - \frac{21S^{2}}{16} + \frac{21}{8}\mu S^{2}, }\\ &{O_{002} = - \frac{3}{4} + \frac{3\mu}{2}, }\\ &{O_{110} = \frac{3}{8}\sqrt{3} C^{2} + \frac{27}{4}CS - \frac{27}{2}\mu CS - \frac{3}{8}\sqrt{3} S^{2}, }\\ &{O_{300} = \frac{37C^{3}}{32} + \frac{75}{32}\sqrt{3} C^{2}S - \frac{75}{16}\sqrt{3} \mu C^{2}S }\\ &{\phantom{O_{300} =}- \frac{123}{32}CS^{2} - \frac{45}{32}\sqrt{3} S^{3} + \frac{45}{16}\sqrt{3} \mu S^{3}, }\\ &{O_{210} = \frac{75}{32}\sqrt{3} C^{3} - \frac{75}{16}\sqrt{3} \mu C^{3} - \frac{357}{32}C^{2}S - \frac{285}{32}\sqrt{3} CS^{2} }\\ &{\phantom{O_{210} =} + \frac{285}{16}\sqrt{3} \mu CS^{2} + \frac{123}{32}S^{3}, }\\ &{O_{030} = - \frac{45}{32}\sqrt{3} C^{3} + \frac{45}{16}\sqrt{3} \mu C^{3} + \frac{123}{32}C^{2}S + \frac{75}{32}\sqrt{3} CS^{2} }\\ &{\phantom{O_{030} =} - \frac{75}{16}\sqrt{3} \mu CS^{2} - \frac{37}{32}S^{3}, }\\ &{O_{102} = \frac{3}{8}C + \frac{15}{8}\sqrt{3} S - \frac{15}{4}\sqrt{3} \mu S, }\\ &{O_{012} = \frac{15}{8} \sqrt{3} C - \frac{15}{4}\sqrt{3}, }\\ &{P_{200} = \frac{3}{16}\sqrt{3} C^{2} - \frac{33}{8}CS - \frac{33}{4}\mu CS + \frac{9}{16}\sqrt{3} S^{2}, }\\ &{P_{020} = \frac{9}{16}\sqrt{3} C^{2} - \frac{33}{8}CS + \frac{33}{4}\mu CS + \frac{3}{16}\sqrt{3} S^{2}, }\\ &{P_{002} = - \frac{3\sqrt{3}}{4}, }\\ &{P_{110} = \frac{33}{8}C^{2} - \frac{33}{4}\mu C^{2} + \frac{3}{4}\sqrt{3} CS - \frac{33}{8}S^{2} + \frac{33}{4}\mu S^{2}, }\\ &{P_{300} = \frac{25}{32}\sqrt{3} C^{3} - \frac{25}{16}\sqrt{3} \mu C^{3} - \frac{123}{32}C^{2}S - \frac{135}{32}\sqrt{3} CS^{2} }\\ &{ \phantom{P_{300} =}+ \frac{135}{16}\sqrt{3} \mu CS^{2} - \frac{3S^{3}}{32}, }\\ &{P_{210} = - \frac{123}{32}C^{3} - \frac{345}{32} \sqrt{3} C^{2}S + \frac{345}{16}\sqrt{3} \mu C^{2}S }\\ &{\phantom{P_{210} =} + \frac{237}{32}CS^{2} + \frac{135}{32}\sqrt{3} S^{3} - \frac{135}{16}\sqrt{3} \mu S^{3}, }\\ &{P_{120} = - \frac{135}{32}\sqrt{3} C^{3} + \frac{135}{16}\sqrt{3} \mu C^{3} + \frac{237}{32}C^{2}S }\\ &{\phantom{P_{120} =}+ \frac{345}{32}\sqrt{3} CS^{2} - \frac{345}{16}\sqrt{3} \mu CS^{2} - \frac{123}{32}S^{3}, }\\ &{P_{030} = - \frac{3C^{3}}{32} + \frac{135}{32}\sqrt{3} C^{2}S - \frac{135}{16}\sqrt{3} \mu C^{2}S }\\ &{\phantom{P_{030} =}- \frac{123}{32}CS^{2} - \frac{25}{32}\sqrt{3} S^{3} + \frac{25}{16}\sqrt{3} \mu S^{3}, }\\ &{P_{102} = \frac{15}{8}\sqrt{3} C - \frac{15}{4} \sqrt{3} \mu C + \frac{33}{8}S, }\\ &{P_{012} = \frac{33}{8}C - \frac{15}{8}\sqrt{3} S + \frac{15}{4}\sqrt{3} \mu S, }\\ &{Q_{101} = \frac{3}{2} + 3\mu,\qquad Q_{011} = - \frac{3\sqrt{3}}{2}, }\\ &{Q_{201} = \frac{3}{8},\qquad Q_{111} = \frac{15\sqrt{3}}{4} + \frac{15\sqrt{3}}{2}\mu, }\\ &{Q_{021} = \frac{33}{8},\qquad Q_{003} = - \frac{3}{2}, }\\ &{O_{101} = 0,\qquad O_{011} = 0,\qquad O_{003} = 0, }\\ &{O_{201} = 0,\qquad O_{021} = 0,\qquad O_{111} = 0, }\\ &{P_{101} = 0,\quad\quad P_{011} = 0,\qquad P_{003} = 0, }\\ &{P_{201} = 0,\qquad P_{021} = 0,\qquad P_{111} = 0, }\\ &{Q_{200} = 0,\qquad Q_{020} = 0,\qquad Q_{002} = 0, }\\ &{Q_{110} = 0,\qquad Q_{300} = 0,\qquad Q_{030} = 0, }\\ &{Q_{210} = 0,\qquad Q_{120} = 0,\qquad Q_{102} = 0, \qquad Q_{012} = 0.} \end{aligned}$$

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Authors and Affiliations

Ying-Jing Qian
- 1
Xiao-Dong Yang
- 1
Email author
Guan-Qiao Zhai
- 1
Wei Zhang
- 1

1.Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical EngineeringBeijing University of TechnologyBeijingP.R. China

Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

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Acknowledgements

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