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Perturbation to Noether symmetries and adiabatic invariants for Birkhoffian systems based on El-Nabulsi dynamical models

Chen Ju Zhang Yi

Perturbation to Noether symmetries and adiabatic invariants for Birkhoffian systems based on El-Nabulsi dynamical models

Chen Ju, Zhang Yi
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  • In this paper, we study the problem of perturbation to Noether symmetries and adiabatic invariants for a Birkhoffian system under small disturbance based on the El-Nabulsi dynamical model. First, the dynamical model presented by El-Nabulsi, which is based on the Riemann-Liouville fractional integral under the framework of the fractional calculus, is extended to the Birkhoffian system, and El-Nabulsi-Birkhoff equations for the Birkhoffian system are established. Then, by using the invariance of the El-Nabulsi-Pfaff action under the infinitesimal transformations, the definition and criterion of the Noether quasi-symmetric transformation are given, and the exact invariant caused directly by the Noether symmetry is obtained. Furthermore, by introducing the concept of high-order adiabatic invariant of a mechanical system, the relationship between the perturbation to the Noether symmetry and the adiabatic invariant after the action of small disturbance is studied, the condition that the perturbation of symmetry leads to the adiabatic invariant and its formulation are presented. As a special case, the perturbation to Noether symmetries and corresponding adiabatic invariants mechanics of non-conservative systems in phase space under El-Nabulsi models and classical Birkhoffian systems are discussed. At the end of the paper, taking the well-known Hojman-Urrutia problem for example, its Noether symmetries under the El-Nabulsi dynamical model is investigated and corresponding exact invariants and adiabatic invariants are presented.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 10972151, 11272227), the Scientific Research and Innovation Program for the Graduate Students in Institution of Higher Education of Jiangsu Province, China (Grant No. CXLX13-855), and the Scientific Research and Innovation Program for the Graduate Students of Suzhou University of Science and Technology, China (Grant No. SKCX13S-050).
    [1]

    Birkhoff G D 1927 Dynamical Systems (Providence: AMS College Publication) pp55-58, 89-96

    [2]

    Santilli R M 1983 Foundations of Theoretical Mechanics (II) (New York: Springer Verlag) pp30-42

    [3]

    Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) pp37-95 (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 BIRKHOFF 系统动力学 (北京: 北京理工大学出版社) 第37–95页]

    [4]

    Galiullan A S 1989 Analytical Dynamics (Moscow: Nauka) pp249-263 (in Russian)

    [5]

    Mei F X 2013 Dynamics of Generalized Birkhoffian System (Beijing: Science Press) pp1-29 (in Chinese) [梅凤翔 2013 广义Birkhoff系统动力学 (北京: 科学出版社) 第1–29页]

    [6]

    Mei F X 1996 Mech. Eng. 18 1 (in Chinese) [梅凤翔 1996 力学与实践 18 1]

    [7]

    Mei F X 1993 Sci. China A 36 1456

    [8]

    Mei F X 2001 Int. J. Non-Linear Mech. 36 817

    [9]

    Guo Y X, Luo S K, Shang M, Mei F X 2001 Rep. Math. Phys. 47 313

    [10]

    Zheng G H, Chen X W, Mei F X 2001 J. Beijing Inst. Technol. 10 17

    [11]

    Zhang Y 2010 Chin. Phys. B 19 080301

    [12]

    Wu H B, Mei F X 2011 Chin. Phys. B 20 104501

    [13]

    Jiang W, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075

    [14]

    Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117

    [15]

    Zhang Y, Mei F X 2004 Acta Phys. Sin. 53 2419 (in Chinese) [张毅, 梅凤翔 2004 物理学报 53 2419]

    [16]

    Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) pp200-226, 459-475 (in Chinese)[梅凤翔 1999 约束力学系统Lie群和Lie代数的应用 (北京: 科学出版社) 第200–226, 459–475页]

    [17]

    Fu J L, Chen L Q 2004 Phys. Lett. A 324 95

    [18]

    Zhang Y 2006 Acta Phys. Sin. 55 3833 (in Chinese) [张毅 2006 物理学报 55 3833]

    [19]

    Zhang H B 2001 Acta Phys. Sin. 50 1837 (in Chinese) [张宏彬 2001 物理学报 50 1837]

    [20]

    Luo S k, Guo Y X 2007 Commun. Theor. Phys. (Beijing) 47 25

    [21]

    El-Nabulsi A R 2005 Fizika A 14 289

    [22]

    El-Nabuls A R 2007 Math. Methods Appl. Sci. 30 1931

    [23]

    El-Nabulsi A R, Torres D F M 2008 J. Math. Phys. 49 053521

    [24]

    El-Nabulsi A R 2009 Chaos Solitons Fract. 42 52

    [25]

    El-Nabulsi A R 2013 Qual. Theory Dyn. Syst. 12 273

    [26]

    Zhang Y 2013 Acta Sci. Nat. Univ. Sunyatseni 52 45 (in Chinese) [张毅 2013 中山大学学报 (自然科学版) 52 45]

    [27]

    Zhang Y 2013 Acta Phys. Sin. 62 164501 (in Chinese) [张毅 2013 物理学报 62 164501]

    [28]

    Long Z X, Zhang Y 2014 Acta Mech. 225 77

    [29]

    Long Z X, Zhang Y 2014 Int. J. Theor. Phys. 53 841

    [30]

    Ding J F, Zhang Y 2014 J. Univ. Sci. Technol. Suzhou (Nat. Sci. Ed.) 31 1 (in Chinese) [丁金凤, 张毅 2014 苏州科技学院学报 (自然科学版) 31 1]

    [31]

    Zhang Y, Zhou Y 2013 Nonlinear Dyn. 73 783

    [32]

    Hojman S, Urrutia L E 1981 J. Math. Phys. 22 1896

    [33]

    Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) p164 (in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与守恒量 (北京: 科学出版社) 第164页]

    [34]

    Zhao Y Y, Mei F X 1996 Acta Mech. Sin. 28 207 (in Chinese) [赵跃宇, 梅凤翔 1996 力学学报 28 207]

  • [1]

    Birkhoff G D 1927 Dynamical Systems (Providence: AMS College Publication) pp55-58, 89-96

    [2]

    Santilli R M 1983 Foundations of Theoretical Mechanics (II) (New York: Springer Verlag) pp30-42

    [3]

    Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) pp37-95 (in Chinese) [梅凤翔, 史荣昌, 张永发, 吴惠彬 1996 BIRKHOFF 系统动力学 (北京: 北京理工大学出版社) 第37–95页]

    [4]

    Galiullan A S 1989 Analytical Dynamics (Moscow: Nauka) pp249-263 (in Russian)

    [5]

    Mei F X 2013 Dynamics of Generalized Birkhoffian System (Beijing: Science Press) pp1-29 (in Chinese) [梅凤翔 2013 广义Birkhoff系统动力学 (北京: 科学出版社) 第1–29页]

    [6]

    Mei F X 1996 Mech. Eng. 18 1 (in Chinese) [梅凤翔 1996 力学与实践 18 1]

    [7]

    Mei F X 1993 Sci. China A 36 1456

    [8]

    Mei F X 2001 Int. J. Non-Linear Mech. 36 817

    [9]

    Guo Y X, Luo S K, Shang M, Mei F X 2001 Rep. Math. Phys. 47 313

    [10]

    Zheng G H, Chen X W, Mei F X 2001 J. Beijing Inst. Technol. 10 17

    [11]

    Zhang Y 2010 Chin. Phys. B 19 080301

    [12]

    Wu H B, Mei F X 2011 Chin. Phys. B 20 104501

    [13]

    Jiang W, Li L, Li Z J, Luo S K 2012 Nonlinear Dyn. 67 1075

    [14]

    Li Z J, Luo S K 2012 Nonlinear Dyn. 70 1117

    [15]

    Zhang Y, Mei F X 2004 Acta Phys. Sin. 53 2419 (in Chinese) [张毅, 梅凤翔 2004 物理学报 53 2419]

    [16]

    Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) pp200-226, 459-475 (in Chinese)[梅凤翔 1999 约束力学系统Lie群和Lie代数的应用 (北京: 科学出版社) 第200–226, 459–475页]

    [17]

    Fu J L, Chen L Q 2004 Phys. Lett. A 324 95

    [18]

    Zhang Y 2006 Acta Phys. Sin. 55 3833 (in Chinese) [张毅 2006 物理学报 55 3833]

    [19]

    Zhang H B 2001 Acta Phys. Sin. 50 1837 (in Chinese) [张宏彬 2001 物理学报 50 1837]

    [20]

    Luo S k, Guo Y X 2007 Commun. Theor. Phys. (Beijing) 47 25

    [21]

    El-Nabulsi A R 2005 Fizika A 14 289

    [22]

    El-Nabuls A R 2007 Math. Methods Appl. Sci. 30 1931

    [23]

    El-Nabulsi A R, Torres D F M 2008 J. Math. Phys. 49 053521

    [24]

    El-Nabulsi A R 2009 Chaos Solitons Fract. 42 52

    [25]

    El-Nabulsi A R 2013 Qual. Theory Dyn. Syst. 12 273

    [26]

    Zhang Y 2013 Acta Sci. Nat. Univ. Sunyatseni 52 45 (in Chinese) [张毅 2013 中山大学学报 (自然科学版) 52 45]

    [27]

    Zhang Y 2013 Acta Phys. Sin. 62 164501 (in Chinese) [张毅 2013 物理学报 62 164501]

    [28]

    Long Z X, Zhang Y 2014 Acta Mech. 225 77

    [29]

    Long Z X, Zhang Y 2014 Int. J. Theor. Phys. 53 841

    [30]

    Ding J F, Zhang Y 2014 J. Univ. Sci. Technol. Suzhou (Nat. Sci. Ed.) 31 1 (in Chinese) [丁金凤, 张毅 2014 苏州科技学院学报 (自然科学版) 31 1]

    [31]

    Zhang Y, Zhou Y 2013 Nonlinear Dyn. 73 783

    [32]

    Hojman S, Urrutia L E 1981 J. Math. Phys. 22 1896

    [33]

    Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press) p164 (in Chinese) [赵跃宇, 梅凤翔 1999 力学系统的对称性与守恒量 (北京: 科学出版社) 第164页]

    [34]

    Zhao Y Y, Mei F X 1996 Acta Mech. Sin. 28 207 (in Chinese) [赵跃宇, 梅凤翔 1996 力学学报 28 207]

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  • Received Date:  27 December 2013
  • Accepted Date:  18 January 2014
  • Published Online:  05 May 2014

Perturbation to Noether symmetries and adiabatic invariants for Birkhoffian systems based on El-Nabulsi dynamical models

  • 1. College of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China;
  • 2. College of Civil Engineering, Suzhou University of Science and Technology, Suzhou 215011, China
Fund Project:  Project supported by the National Natural Science Foundation of China (Grant Nos. 10972151, 11272227), the Scientific Research and Innovation Program for the Graduate Students in Institution of Higher Education of Jiangsu Province, China (Grant No. CXLX13-855), and the Scientific Research and Innovation Program for the Graduate Students of Suzhou University of Science and Technology, China (Grant No. SKCX13S-050).

Abstract: In this paper, we study the problem of perturbation to Noether symmetries and adiabatic invariants for a Birkhoffian system under small disturbance based on the El-Nabulsi dynamical model. First, the dynamical model presented by El-Nabulsi, which is based on the Riemann-Liouville fractional integral under the framework of the fractional calculus, is extended to the Birkhoffian system, and El-Nabulsi-Birkhoff equations for the Birkhoffian system are established. Then, by using the invariance of the El-Nabulsi-Pfaff action under the infinitesimal transformations, the definition and criterion of the Noether quasi-symmetric transformation are given, and the exact invariant caused directly by the Noether symmetry is obtained. Furthermore, by introducing the concept of high-order adiabatic invariant of a mechanical system, the relationship between the perturbation to the Noether symmetry and the adiabatic invariant after the action of small disturbance is studied, the condition that the perturbation of symmetry leads to the adiabatic invariant and its formulation are presented. As a special case, the perturbation to Noether symmetries and corresponding adiabatic invariants mechanics of non-conservative systems in phase space under El-Nabulsi models and classical Birkhoffian systems are discussed. At the end of the paper, taking the well-known Hojman-Urrutia problem for example, its Noether symmetries under the El-Nabulsi dynamical model is investigated and corresponding exact invariants and adiabatic invariants are presented.

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