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

Zhang Hong-Bin Lü Hong-Sheng Gu Shu-Long

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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  • 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.
    • Funds:
    [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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  • Received Date:  31 August 2009
  • Accepted Date:  05 November 2009
  • Published Online:  05 April 2010

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

  • 1. (1)Department of Mathematics, Chaohu College, Chaohu 238000,China; (2)Department of Physics, Chaohu College, Chaohu 238000,China

Abstract: 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.

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