Symplectic Group of the Transfer Matrix Converges to the Symplectic Lie Group

ZHONG Wan-xie; GAO Qiang

doi:10.3879/j.issn.1000-0887.2013.06.001

Volume 34 Issue 6

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2013 > 34(6): 547-551

ZHONG Wan-xie, GAO Qiang. Symplectic Group of the Transfer Matrix Converges to the Symplectic Lie Group[J]. Applied Mathematics and Mechanics, 2013, 34(6): 547-551. doi: 10.3879/j.issn.1000-0887.2013.06.001

Citation:

PDF( 330 KB)

Symplectic Group of the Transfer Matrix Converges to the Symplectic Lie Group

doi: 10.3879/j.issn.1000-0887.2013.06.001

State Key Laboratory of Structural Analysis of Industrial Equipment,Department of Engineering Mechanics,Dalian University of Technology,Dalian, Liaoning 116023, P.R.China

Received Date: 2013-04-24
Rev Recd Date: 2013-05-09
Publish Date: 2013-06-15

Abstract

Abstract

By using action variational principle, the transfer symplectic matrix for the discrete integral of the Hamiltonian canonical equation was given. Then the Lie algebra corresponding to the Hamiltonian canonical equation was given. When the time step tends to zero, that the symplectic group of the transfer matrix for discrete integrator converges to the symplectic Lie group of the continuoustime differential equation of the Hamiltonian system was proved.
- discrete integration,
- symplectic group of the transfer matrix,
- Hamilton,
- symplectic Lie group

FullText(HTML)

References(12)

References

[1]	Goldstein H.Classical Mechanics[M]. 3rd ed. London: AddisonWesley, 2002.
[2]	Arnold V I.Mathematical Methods of Classical Mechanics[M]. New York: SpringerVerlag, 1989.
[3]	冯康, 秦孟兆. 哈密尔顿系统的辛几何算法[M]. 杭州：浙江科学技术出版社，2003. (FENG Kang, QIN Meng-zhao.Symplectic Geometric Algorithms for Hamiltonian Systems[M]. Hangzhou：Zhejiang Science & Technology Press, 2003.(in Chinese)）
[4]	Hairer E, Lubich C, Wanner G.Geometric Numerical Integration: Structure-Preserving Algorithm for Ordinary Differential Equations[M]. New York: Springer, 2006.
[5]	Hairer E, Norsett S P, Wanner G.Solving Ordinary Differential Equations INonstiff Problem[M]. 2nd ed. Berlin: Springer, 1993.
[6]	Hairer E, Lubich C, Wanner G.Geometric Numerical Integration: Structure-Preserving Algorithm for Ordinary Differential Equations[M]. 2nd ed. New York: Springer, 2006.
[7]	钟万勰.分析结构力学与有限元[J]. 动力学与控制学报, 2004, 2(4)：18.（ZHONG Wan-xie. Analytical structural mechanics and finite element[J].Journal of Dynamics and Control, 2004, 2(4)：18.(in Chinese)）
[8]	钟万勰, 姚征. 时间有限元与保辛[J]. 机械强度, 2005, 27(2)：178183.(ZHONG Wan-xie, YAO Zheng. Time domain FEM and symplectic conservation[J].Journal of Mechanical Strength ，2005, 27(2)：178183.(in Chinese))
[9]	钟万勰, 高强.约束动力系统的分析结构力学积分[J]. 动力学与控制, 2006, 4(3)：193200.(ZHONG Wan-xie, GAO Qiang. Integration of constrained dynamical system via analytical structural mechanics[J]. Journal of Dynamics and Control， 2006, 4(3)：193200.(in Chinese))
[10]	钟万勰, 高强.辛破茧[M].大连：大连理工大学出版社, 2011.(ZHONG Wan-xie, GAO Qiang.Break the Limitation of Symplecticity[M]. Dalian: Dalian University of Technology Press, 2011.(in Chinese))
[11]	钟万勰. 力、功、能量与辛数学[M]. 大连：大连理工大学出版社, 2007.（ZHONG Wan-xie.Force, Work, Energy and Sympletic Mathematics[M]. Dalian: Dalian University of Technology Press, 2007.(in Chinese)）
[12]	Gao Q, Tan S J, Zhang H W, Zhong W X. Symplectic algorithms based on the principle of least action and generating functions[J].International Journal for Numerical Methods in Engineering, 2012, 89(4): 438508.