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旋转坐标系下的圆型限制性三体问 题因含非惯性系所附加的影响部分使得动能不是动量的严格二次型, 可能导致力梯度辛积分算法的应用遇到困难. 从Lie算子运算出发, 严格论证了力梯度算子在这种情形下的物理意义 仍然像质心惯性坐标系下的圆型限制性三体问题那样是引力的梯度, 而不是引力与非惯性力所得合力的梯度, 表明了力梯度辛方法适合求解旋转坐标系下的圆型限制性三体问题. 通过应用四阶力梯度辛方法、最优化四阶力梯度辛方法和Forest-Ruth 辛方法分别求解该问题, 进行了数值对比研究, 结果显示最优化型力梯度算法能够取得最好精度. 还应用最优化型算法计算两邻近轨道的Lyapunov指数和快速Lyapunov指标, 确保高精度辛方法能够贯穿于这些混沌指标计算的全过程, 以便准确刻画此系统的动力学定性性质.
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关键词:
- 辛积分器 /
- 圆型限制性三体问题 /
- 混沌 /
- Lyapunov指数
The kinetic energy of the circular restricted three-body problem in a rotating frame is no longer a standard positive quadratic function of moment, owing to the additional part in the non-inertial rotating frame, which leads to a difficulty in using force gradient symplectic integrators. To address this problem, we show through the calculation of Lie operators that the force gradient operator on the system is still related to the gradient of the gravitational forces from the two main objects rather than that of the resultant force of both the gravitational forces and the non-inertial force exerted by the rotating frame, just as the force gradient operator on the circular restricted three-body problem in an inertial frame. Therefore, it is reasonable to use the gradient symplectic integrators for integrating the circular restricted three-body problem in the rotating frame from a theoretical point of view. Numerical simulations describe that a fourth-order force gradient symplectic method is always greatly superior to the non-gradient Forest-Ruth algorithm in the numerical accuracy, and its optimized version is best. Because of this, the optimized gradient scheme is recommended for calculating chaos indicators, such as Lyapunov exponents of and fast Lyapunov indicators of two nearby trajectories, which is conductive to obtaining a true description of dynamically qualitative properties.-
Keywords:
- symplectic integrators /
- circular restricted three-body problem /
- chaos /
- Lyapunov exponents
[1] Feng K, Qin M Z 2009 Symplectic Geometric Algorithms for Hamiltonian Systems (Hangzhou: Zhejiang Science and Technology Publishing House)
[2] Zhong S Y, Wu X 2010 Phys. Rev. D 81 104037
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[15] Ruth R D 1983 IEEE Tran. Nucl. Sci. 30 2669
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[17] Liao X H 1997 Celest. Mech. Dyn. Astron. 66 243
[18] Lubich C, Walther B, Braugmann B 2010 Phys. Rev. D 81 104025
[19] Forest E, Ruth R D 1990 Physica D 43 105
[20] Yoshida H 1990 Phys. Lett. A 150 262
[21] Wisdom J, Holman M 1991 Astron. J. 102 1528
[22] Preto M, Tremaine S 1999 Astron. J. 118 2532
[23] Laskar J, Robutel P 2001 Celest. Mech. Dyn. Astron. 80 39
[24] Wisdom J, Holman M, Touma J 1996 Fields Inst. Commun. 10 217
[25] Chin S A 1997 Phys. Lett. A 75 226
[26] Chin S A, Chen C R 2005 Celest. Mech. Dyn. Astron. 91 301
[27] Chin S A 2007 Phys. Rev. E 75 036701
[28] Xu J, Wu X 2010 Res. Astron. Astrophys. 10 173
[29] Sun W, Wu X, Huang G Q 2011 Res. Astron. Astrophys. 11 353
[30] Li R, Wu X 2010 Science China: Physics, Mechanics & Astronomy 53 1600
[31] Omelyan I P, Mryglod I M, Folk R 2003 Comput. Phys. Commun. 151 272
[32] Li R, Wu X 2010 Acta Phys. Sin. 59 7135 (in Chinese) [李荣,伍歆 2010 物理学报 59 7135]
[33] Wu X, Huang T Y 2003 Phys. Lett. A 313 77
[34] Wu X, Huang T Y, Zhang H 2006 Phys. Rev. D 74 083001
[35] Murray C D, Dermott S F 1999 Solar System Dynamics (Cambridge, UK: Cambridge Univ. Press)
[36] Tancredi G, Sánchez A, Roig F 2001 Astron. J. 121 1171
[37] Froeschlé C, Lega E 2000 Celest. Mech. Dyn. Astron. 78 167
[38] Wu X, Xie Y 2008 Phys. Rev. D 77 103012
[39] Wang Y, Wu X 2012 Chin. Phys. B 21 050504
[40] Wang Y Z, Wu X, Zhong S Y 2012 Acta Phys. Sin. 61 160401 (in Chinese) [王玉诏, 伍歆, 钟双英 2012 物理学报 61 160401]
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[1] Feng K, Qin M Z 2009 Symplectic Geometric Algorithms for Hamiltonian Systems (Hangzhou: Zhejiang Science and Technology Publishing House)
[2] Zhong S Y, Wu X 2010 Phys. Rev. D 81 104037
[3] Mei L J, Wu X, Liu F Y 2012 Chin. Phys. Lett. 29 050201
[4] Hairer E, Lubich C, Wanner G 1999 Geometric Numerical Integration. (Berlin: Springer)
[5] Chi Y H, Liu X S, Ding P Z 2006 Acta Phys. Sin. 55 6320 (in Chinese) [匙玉华, 刘学深, 丁培柱 2006 物理学报 55 6320]
[6] Luo X Y, Liu X S, Ding P Z 2007 Acta Phys. Sin. 56 604 (in Chinese) [罗香怡, 刘学深, 丁培柱 2007 物理学报 56 604]
[7] Liu X S, Wei J Y, Ding P Z 2005 Chin. Phys. 14 231
[8] Bian X B, Qiao H X, Shi T Y 2007 Chin. Phys. 16 1822
[9] Cao Y, Yang K Q 2003 Acta Phys. Sin. 52 1984 (in Chinese) [曹禹, 杨孔庆 2003 物理学报 52 1984]
[10] Hu W P, Deng Z C 2008 Chin. Phys. B 17 3923
[11] Zhong S Y, Wu X, Liu S Q, Deng X F 2010 Phys. Rev. D 82 124040
[12] Zhong S Y, Wu X 2011 Acta Phys. Sin. 60 090402 (in Chinese) [钟双英, 伍歆 2011 物理学报 60 090402]
[13] Zhong S Y, Liu S 2012 Acta Phys. Sin. 61 120401 (in Chinese) [钟双英, 刘崧 2012 物理学报 61 120401]
[14] Wu X, Zhong S Y 2011 Gen. Relat. Gravit. 43 2185
[15] Ruth R D 1983 IEEE Tran. Nucl. Sci. 30 2669
[16] Preto M, Saha P 2009 Astrophy. J. 703 1743
[17] Liao X H 1997 Celest. Mech. Dyn. Astron. 66 243
[18] Lubich C, Walther B, Braugmann B 2010 Phys. Rev. D 81 104025
[19] Forest E, Ruth R D 1990 Physica D 43 105
[20] Yoshida H 1990 Phys. Lett. A 150 262
[21] Wisdom J, Holman M 1991 Astron. J. 102 1528
[22] Preto M, Tremaine S 1999 Astron. J. 118 2532
[23] Laskar J, Robutel P 2001 Celest. Mech. Dyn. Astron. 80 39
[24] Wisdom J, Holman M, Touma J 1996 Fields Inst. Commun. 10 217
[25] Chin S A 1997 Phys. Lett. A 75 226
[26] Chin S A, Chen C R 2005 Celest. Mech. Dyn. Astron. 91 301
[27] Chin S A 2007 Phys. Rev. E 75 036701
[28] Xu J, Wu X 2010 Res. Astron. Astrophys. 10 173
[29] Sun W, Wu X, Huang G Q 2011 Res. Astron. Astrophys. 11 353
[30] Li R, Wu X 2010 Science China: Physics, Mechanics & Astronomy 53 1600
[31] Omelyan I P, Mryglod I M, Folk R 2003 Comput. Phys. Commun. 151 272
[32] Li R, Wu X 2010 Acta Phys. Sin. 59 7135 (in Chinese) [李荣,伍歆 2010 物理学报 59 7135]
[33] Wu X, Huang T Y 2003 Phys. Lett. A 313 77
[34] Wu X, Huang T Y, Zhang H 2006 Phys. Rev. D 74 083001
[35] Murray C D, Dermott S F 1999 Solar System Dynamics (Cambridge, UK: Cambridge Univ. Press)
[36] Tancredi G, Sánchez A, Roig F 2001 Astron. J. 121 1171
[37] Froeschlé C, Lega E 2000 Celest. Mech. Dyn. Astron. 78 167
[38] Wu X, Xie Y 2008 Phys. Rev. D 77 103012
[39] Wang Y, Wu X 2012 Chin. Phys. B 21 050504
[40] Wang Y Z, Wu X, Zhong S Y 2012 Acta Phys. Sin. 61 160401 (in Chinese) [王玉诏, 伍歆, 钟双英 2012 物理学报 61 160401]
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