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完整约束力学系统保Lie对称性差分格式

张宏彬 吕洪升 顾书龙

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完整约束力学系统保Lie对称性差分格式

张宏彬, 吕洪升, 顾书龙

The Lie point symmetry-preserving difference scheme of holonomic constrained mechanical systems

Zhang Hong-Bin, Lü Hong-Sheng, Gu Shu-Long
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  • 提出一种保完整约束力学系统Lie点对称性的差分格式.其具体做法是:首先求出完整约束力学系统的Lie点对称群,并将其延拓到离散的差分格点上;其次利用其特征方程找出独立的离散不变量;再应用独立的离散不变量来构造不变量的差分格式,且此差分格式在连续极限下应给出原系统的微分方程;最后给出一个例子,作为本文结果的说明.
    In this paper a Lie point symmetry- preserving difference schemes approximating holonomic constrained mechanical systems is presented. The procedure is as follows: Firstly we find the Lie point symmetry groups of the original equation, and prolong them to three points of the lattice. Secondly the discrete invariants are obtained by solving discrete characteristic equations, then the invariant difference scheme is constructed by using these discrete invariants; and this invariants difference scheme will give the original equation under the continuous limit. Finally an example is presented to illustrate the applications of the result.
    • 基金项目: 国家自然科学基金(批准号:10872037),安徽省自然科学基金(批准号:070416226)资助的课题.
    [1]

    Lie S 1889 Die infinitesimalen Beruhrungstransformationen der Mechanik (Leipz: Berichte)

    [2]

    Olver P J 1986 Applications of Lie Groups to differential Equations ( New York: Springer)

    [3]

    Wluman G and Kumei S 1989 Symmetries and Differential Equations (Berlin: Springer)

    [4]

    Ibragimov N H 1985 Transformation Groups Applied to Mathematical Physics ( Boston: Reidel)

    [5]

    Noether A E 1918 Nachr. Akad. Wiss. Gottingen Math.Phys. KI II 235

    [6]

    Li Z P 1981 Acta Phys.Sin 30 1659 (in Chinese) [李子平 1981 物理学报30 1659]

    [7]

    Luo Y, Zhao Y Y 1986 J.Beijing Inst. Technol 6 41 (in Chinese) [罗 勇、赵跃宇 1986 北京工业学院学报 6 41]

    [8]

    Zhao Y Y, Mei F X 1999 Symmetries and Conserved quantities of Mechanical systems (Beijing: Science Press) (in Chinese) [赵跃宇、梅凤翔 1999 (北京:科学出版社)]

    [9]

    Mei F X 1999 Applications of Lie Groups and Lie algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)[梅凤翔 1999 李群和李代数对约束力学系统应用 (北京:科学出版社)]

    [10]

    Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量 (北京:北京理工大学出版社)]

    [11]

    Levi D, Winternitz P 1991 Phys.Lett. 152 A 335

    [12]

    Levi D, Winternitz P 1993 J.Math.Phys. 34 3713

    [13]

    Levi D, Winternitz P 1995 Symmetries of Discrete Dynamical Systems. Technical Report CRM-2312. 1995, Centre de recherches mathématiques,Université de Motréal

    [14]

    Floreanini R, Negro J, Nieto L M, Vinet L 1996 Lett.Math.Phys. 36 351

    [15]

    Floreanini R, Vinet L 1995 J.Math.Phys. 36 7024

    [16]

    Hernández Heredero R, Levi D 2003 J.Nonl.Math.Phys. 10 Suppl 2 77

    [17]

    Hernández Heredero R, Levi D, M A Rodriguez, P Winternitz 2000 J.Phys.A:Math.Gen. 33 5025

    [18]

    Hernández Heredero R, Levi D, Rodriguez M A, Winternitz P 2001 J.Phys.A:Math.Gen. 34 2459

    [19]

    Dorodnitsyn V A 1991 J.Sov.Math. 55 1490

    [20]

    Dorodnitsyn V A 1993 Dokl.Ak.Nauk. 328 678

    [21]

    Dorodnitsyn V A 1994 Int.J.Mod.Phys. C5 723

    [22]

    Dorodnitsyn V A, Kozlov R 2003 J.Nonl.Math.Phys. 10 16

    [23]

    Dorodnitsyn V A, Kozlov R, Winternitz P 2000 J.Math.Phys. 41 480

    [24]

    Dorodnitsyn V A, Kozlov R, Winternitz.P 2004 J.Math.Phys. 45 336

    [25]

    Dorodnitsyn V A, Winternitz P 2000 Nonlinear Dynamics. 22 49

    [26]

    Fu J L, Chen L Q, Salnalor J, Tang Y F 2006 Phys. Lett. A 358 5

    [27]

    Fu J L, Dai G D, Jiménes S, Tang Y F 2007 Chin.Phys. 16 570

    [28]

    Fu J L, Chen B Y,Tang Y F, Fu H 2008 Chin.Phys. B 17 3942

    [29]

    Fu J L, Chen B Y, Xie F P 2008 Chin.Phys. B 17 4354

    [30]

    Fu J L, Nie N M, Huang J F, Salvador J, Tang Y F, Lius V, Zhao W J 2009 Chin.Phys. B 18 2634

    [31]

    Fu J L, Chen L Q, Chen B Y 2009 Sci.China G 39 1320

    [32]

    Liu R W, Zhang H B, Chen L Q 2006 Chin.Phys. 15 249

    [33]

    Shi S Y, Fu J L, Chen L Q 2008 Chin. Phys. B 17 385

    [34]

    Shi S Y, Fu J L, Huang X H, Chen L Q, Zhang X B 2008 Chin. Phys. B 17 754

    [35]

    Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin.Phys. 16 582

    [36]

    Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 238

    [37]

    Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 888

    [38]

    Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 1031

  • [1]

    Lie S 1889 Die infinitesimalen Beruhrungstransformationen der Mechanik (Leipz: Berichte)

    [2]

    Olver P J 1986 Applications of Lie Groups to differential Equations ( New York: Springer)

    [3]

    Wluman G and Kumei S 1989 Symmetries and Differential Equations (Berlin: Springer)

    [4]

    Ibragimov N H 1985 Transformation Groups Applied to Mathematical Physics ( Boston: Reidel)

    [5]

    Noether A E 1918 Nachr. Akad. Wiss. Gottingen Math.Phys. KI II 235

    [6]

    Li Z P 1981 Acta Phys.Sin 30 1659 (in Chinese) [李子平 1981 物理学报30 1659]

    [7]

    Luo Y, Zhao Y Y 1986 J.Beijing Inst. Technol 6 41 (in Chinese) [罗 勇、赵跃宇 1986 北京工业学院学报 6 41]

    [8]

    Zhao Y Y, Mei F X 1999 Symmetries and Conserved quantities of Mechanical systems (Beijing: Science Press) (in Chinese) [赵跃宇、梅凤翔 1999 (北京:科学出版社)]

    [9]

    Mei F X 1999 Applications of Lie Groups and Lie algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese)[梅凤翔 1999 李群和李代数对约束力学系统应用 (北京:科学出版社)]

    [10]

    Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量 (北京:北京理工大学出版社)]

    [11]

    Levi D, Winternitz P 1991 Phys.Lett. 152 A 335

    [12]

    Levi D, Winternitz P 1993 J.Math.Phys. 34 3713

    [13]

    Levi D, Winternitz P 1995 Symmetries of Discrete Dynamical Systems. Technical Report CRM-2312. 1995, Centre de recherches mathématiques,Université de Motréal

    [14]

    Floreanini R, Negro J, Nieto L M, Vinet L 1996 Lett.Math.Phys. 36 351

    [15]

    Floreanini R, Vinet L 1995 J.Math.Phys. 36 7024

    [16]

    Hernández Heredero R, Levi D 2003 J.Nonl.Math.Phys. 10 Suppl 2 77

    [17]

    Hernández Heredero R, Levi D, M A Rodriguez, P Winternitz 2000 J.Phys.A:Math.Gen. 33 5025

    [18]

    Hernández Heredero R, Levi D, Rodriguez M A, Winternitz P 2001 J.Phys.A:Math.Gen. 34 2459

    [19]

    Dorodnitsyn V A 1991 J.Sov.Math. 55 1490

    [20]

    Dorodnitsyn V A 1993 Dokl.Ak.Nauk. 328 678

    [21]

    Dorodnitsyn V A 1994 Int.J.Mod.Phys. C5 723

    [22]

    Dorodnitsyn V A, Kozlov R 2003 J.Nonl.Math.Phys. 10 16

    [23]

    Dorodnitsyn V A, Kozlov R, Winternitz P 2000 J.Math.Phys. 41 480

    [24]

    Dorodnitsyn V A, Kozlov R, Winternitz.P 2004 J.Math.Phys. 45 336

    [25]

    Dorodnitsyn V A, Winternitz P 2000 Nonlinear Dynamics. 22 49

    [26]

    Fu J L, Chen L Q, Salnalor J, Tang Y F 2006 Phys. Lett. A 358 5

    [27]

    Fu J L, Dai G D, Jiménes S, Tang Y F 2007 Chin.Phys. 16 570

    [28]

    Fu J L, Chen B Y,Tang Y F, Fu H 2008 Chin.Phys. B 17 3942

    [29]

    Fu J L, Chen B Y, Xie F P 2008 Chin.Phys. B 17 4354

    [30]

    Fu J L, Nie N M, Huang J F, Salvador J, Tang Y F, Lius V, Zhao W J 2009 Chin.Phys. B 18 2634

    [31]

    Fu J L, Chen L Q, Chen B Y 2009 Sci.China G 39 1320

    [32]

    Liu R W, Zhang H B, Chen L Q 2006 Chin.Phys. 15 249

    [33]

    Shi S Y, Fu J L, Chen L Q 2008 Chin. Phys. B 17 385

    [34]

    Shi S Y, Fu J L, Huang X H, Chen L Q, Zhang X B 2008 Chin. Phys. B 17 754

    [35]

    Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin.Phys. 16 582

    [36]

    Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 238

    [37]

    Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 888

    [38]

    Zhang H B, Chen L Q, Liu R W 2005 Chin. Phys. 14 1031

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出版历程
  • 收稿日期:  2009-08-31
  • 修回日期:  2009-11-05
  • 刊出日期:  2010-04-05

完整约束力学系统保Lie对称性差分格式

  • 1. (1)巢湖学院数学系,巢湖 238000; (2)巢湖学院物理系,巢湖 238000
    基金项目: 国家自然科学基金(批准号:10872037),安徽省自然科学基金(批准号:070416226)资助的课题.

摘要: 提出一种保完整约束力学系统Lie点对称性的差分格式.其具体做法是:首先求出完整约束力学系统的Lie点对称群,并将其延拓到离散的差分格点上;其次利用其特征方程找出独立的离散不变量;再应用独立的离散不变量来构造不变量的差分格式,且此差分格式在连续极限下应给出原系统的微分方程;最后给出一个例子,作为本文结果的说明.

English Abstract

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