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

Chen Ju; Zhang Yi

doi:10.7498/aps.63.104501

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.

Keywords:

Authors and contacts

Chen Ju¹,
Zhang Yi²

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
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).

References

[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]

Cited By

[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]	Zhang Yi. Mei’s symmetry theorems for non-migrated Birkhoffian systems on a time scale. Acta Physica Sinica, 2021, 70(24): 244501. doi: 10.7498/aps.70.20210372
[2]	Cui Jin-Chao, Liao Cui-Cui, Zhao Zhe, Liu Shi-Xing. A simplified method of solving Birkhoffian function and Lagrangian. Acta Physica Sinica, 2016, 65(18): 180201. doi: 10.7498/aps.65.180201
[3]	Chen Ju, Zhang Yi. Exact invariants and adiabatic invariants for nonholonomic systems in non-Chetaev's type based on El-Nabulsi dynamical models. Acta Physica Sinica, 2015, 64(3): 034502. doi: 10.7498/aps.64.034502
[4]	Zhang Yi. Perturbation to Noether symmetries and adiabatic invariants for nonconservative dynamic systems. Acta Physica Sinica, 2013, 62(16): 164501. doi: 10.7498/aps.62.164501
[5]	Ding Guang-Tao. Hojman method for construction of Birkhoffian representation and the Birkhoff symmetry. Acta Physica Sinica, 2010, 59(6): 3643-3647. doi: 10.7498/aps.59.3643
[6]	Ding Guang-Tao. Effects of gauge transformations on symmetries of Birkhoffian system. Acta Physica Sinica, 2009, 58(11): 7431-7435. doi: 10.7498/aps.58.7431
[7]	Zhang Yi. Noether’s theory for Birkhoffian systems in the event space. Acta Physica Sinica, 2008, 57(5): 2643-2648. doi: 10.7498/aps.57.2643
[8]	Jing Hong-Xing, Li Yuan-Cheng, Xia Li-Li. Perturbation of Lie symmetries and a type of generalized Hojman adiabatic invariants for variable mass systems with unilateral holonomic constraints. Acta Physica Sinica, 2007, 56(6): 3043-3049. doi: 10.7498/aps.56.3043
[9]	Zhang Yi. Lie symmetries and adiabatic invariants for holonomic systems in event space. Acta Physica Sinica, 2007, 56(6): 3054-3059. doi: 10.7498/aps.56.3054
[10]	Xia Li-Li, Li Yuan-Cheng. Perturbation to symmetries and adiabatic invariant for nonholonomic controllable mechanical system in phase place. Acta Physica Sinica, 2007, 56(11): 6183-6187. doi: 10.7498/aps.56.6183
[11]	Zhang Yi. Perturbation of symmetries and Hojman adiabatic invariants of discrete mechanical systems in the phase space. Acta Physica Sinica, 2007, 56(4): 1855-1859. doi: 10.7498/aps.56.1855
[12]	Zhang Peng-Yu, Fang Jian-Hui. Lie symmetry and non-Noether conserved quantities of variable mass Birkhoffian system. Acta Physica Sinica, 2006, 55(8): 3813-3816. doi: 10.7498/aps.55.3813
[13]	Zhang Yi, Fan Cun-Xin, Mei Feng-Xiang. Perturbation of symmetries and Hojman adiabatic invariants for Lagrangian system. Acta Physica Sinica, 2006, 55(7): 3237-3240. doi: 10.7498/aps.55.3237
[14]	Zhang Yi. A new type of adiabatic invariants for Birkhoffian system. Acta Physica Sinica, 2006, 55(8): 3833-3837. doi: 10.7498/aps.55.3833
[15]	Zhang Yi, Mei Feng-Xiang. Effects of constraints on Noether symmetries and conserved quantities in a Birkhoffian system. Acta Physica Sinica, 2004, 53(8): 2419-2423. doi: 10.7498/aps.53.2419
[16]	Fu Jing-Li, Chen Li-Qun, Xie Feng-Ping. Perturbation to the symmetries of relativistic Birkhoffian systems and the inver se problems. Acta Physica Sinica, 2003, 52(11): 2664-2670. doi: 10.7498/aps.52.2664
[17]	Zhang Yi. . Acta Physica Sinica, 2002, 51(3): 461-464. doi: 10.7498/aps.51.461
[18]	Luo Shao-Kai, Lu Yi-Bing, Zhou Qiang, Wang Ying-De, Oyang Shi. . Acta Physica Sinica, 2002, 51(9): 1913-1917. doi: 10.7498/aps.51.1913
[19]	Zhang Yi. . Acta Physica Sinica, 2002, 51(8): 1666-1670. doi: 10.7498/aps.51.1666
[20]	FU JING-LI, CHEN LI-QUN, LUO SHAO-KAI, CHEN XIANG-WEI, WANG XIN-MIN. STUDY ON DYNAMICS OF RELATIVISTIC BIRKHOFF SYSTEMS. Acta Physica Sinica, 2001, 50(12): 2289-2295. doi: 10.7498/aps.50.2289

Catalog

Vol. 63, Issue 10 - 2014-05-20

Metrics

Abstract views: 4904
PDF Downloads: 480
Cited By: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

Search

Article

留言板