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In this paper, the problem of exact invariants and adiabatic invariants for nonholonomic system in non-Chetaev's type based on the El-Nabulsi dynamical model is studied. First, the El-Nabulsi-d'Alembert-Lagrange principle is deduced and the differential equations of motion of the system are established. Then, the relation between the Noether symmetry and the exact invariant that is led directly by the symmetry for undisturbed nonholonomic system in non-Chetaev's type is given. Furthermore, by introducing the concept of high-order adiabatic invariant of a mechanical system, the conditions that the perturbation of symmetry leads to the adiabatic invariant and its formulation are studied for the nonholonomic system in non-Chetaev's type under the action of small disturbance. As a special case, the problem of the exact invariants and the adiabatic invariants for the nonholonomic system in Chetaev's type in El-Nabulsi model is discussed. At the end of the paper, two examples for the nonholonomic systems in non-Chetaev's type constraints and also the Chetaev's type constraints are given respectively to show the application of the methods and the results of this paper.
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Keywords:
- perturbation of symmetry /
- adiabatic invariant /
- nonholonomic system in non-Chetaev' /
- s type
[1] Mei F X 1987 Researches on Nonholonomic Dynamics (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔1987非完整动力学研究(北京: 北京工业学院出版社)]
[2] Bugers J M 1917 Ann. Phys. 357 195
[3] Djuki D S 1981 Int. J. Non-Linear Mech. 16 489
[4] Bulanov S V, Shasharina S G 1992 Nucl. Fusion 32 1531
[5] Notte J, Fajans J, Chu R, Wurtele J S 1993 Phys. Rev. Lett. 70 3900
[6] 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]
[7] 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]
[8] Chen X W, Wang X M, Wang M Q 2004 Chin. Phys. 13 2003
[9] Fu J L, Chen L Q 2004 Phys. Lett. A 324 95
[10] Qiao Y F, Li R J, Sun D N 2005 Chin. Phys. 14 1919
[11] Chen X W, Li Y M 2005 Chin. Phys. 14 663
[12] Chen X W, Liu C M, Li Y M 2006 Chin. Phys. 15 470
[13] Luo S K, Chen X W, Guo Y X 2007 Chin. Phys. 16 3176
[14] Luo S k, Guo Y X 2007 Commun. Theor. Phys. 47 25
[15] Ding N, Fang J H 2009 Acta Phys. Sin. 58 7440 (in Chinese) [丁宁, 方建会 2009 物理学报 58 7440]
[16] El-Nabulsi A R 2005 Fizika A 14 289
[17] El-Nabulsi A R 2005 Int. J. Appl. Math. 17 299
[18] El-Nabulsi A R and Torres D F M 2008 J. Math. Phys. 49 053521
[19] El-Nabuls A R 2007 Math. Methods Appl. Sci. 30 1931
[20] El-Nabulsi A R 2009 Chaos Sol. Fract. 42 52
[21] El-Nabulsi A R, Dzenite A I, Torres D F M 2006 Bound Field Compu. Simu. 48 189
[22] El-Nabulsi A R 2013 Qual. Theory Dyn. Syst. 12 273
[23] El-Nabulsi A R 2007 Rom. J. Phys. 52 705
[24] El-Nabulsi A R 2007 Rom. Rep. Phys. 59 759
[25] Zhang Y 2013 Acta Phys. Sin. 62 164501 (in Chinese) [张毅 2013 物理学报 62 164501]
[26] Chen J, Zhang Y 2014 Acta Phys. Sin. 63 104501 (in Chinese) [陈菊, 张毅 2014 物理学报 63 104501]
[27] Chen J, Zhang Y 2014 Nonlinear Dyn. 77 353
[28] Mei F X 1985 Foundations of Mechanics of Nonholonomic Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔1985非完整系统力学基础(北京: 北京工业学院出版社)]
[29] Mei F X, Wu H B 2009 Dynamics of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press)
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[1] Mei F X 1987 Researches on Nonholonomic Dynamics (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔1987非完整动力学研究(北京: 北京工业学院出版社)]
[2] Bugers J M 1917 Ann. Phys. 357 195
[3] Djuki D S 1981 Int. J. Non-Linear Mech. 16 489
[4] Bulanov S V, Shasharina S G 1992 Nucl. Fusion 32 1531
[5] Notte J, Fajans J, Chu R, Wurtele J S 1993 Phys. Rev. Lett. 70 3900
[6] 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]
[7] 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]
[8] Chen X W, Wang X M, Wang M Q 2004 Chin. Phys. 13 2003
[9] Fu J L, Chen L Q 2004 Phys. Lett. A 324 95
[10] Qiao Y F, Li R J, Sun D N 2005 Chin. Phys. 14 1919
[11] Chen X W, Li Y M 2005 Chin. Phys. 14 663
[12] Chen X W, Liu C M, Li Y M 2006 Chin. Phys. 15 470
[13] Luo S K, Chen X W, Guo Y X 2007 Chin. Phys. 16 3176
[14] Luo S k, Guo Y X 2007 Commun. Theor. Phys. 47 25
[15] Ding N, Fang J H 2009 Acta Phys. Sin. 58 7440 (in Chinese) [丁宁, 方建会 2009 物理学报 58 7440]
[16] El-Nabulsi A R 2005 Fizika A 14 289
[17] El-Nabulsi A R 2005 Int. J. Appl. Math. 17 299
[18] El-Nabulsi A R and Torres D F M 2008 J. Math. Phys. 49 053521
[19] El-Nabuls A R 2007 Math. Methods Appl. Sci. 30 1931
[20] El-Nabulsi A R 2009 Chaos Sol. Fract. 42 52
[21] El-Nabulsi A R, Dzenite A I, Torres D F M 2006 Bound Field Compu. Simu. 48 189
[22] El-Nabulsi A R 2013 Qual. Theory Dyn. Syst. 12 273
[23] El-Nabulsi A R 2007 Rom. J. Phys. 52 705
[24] El-Nabulsi A R 2007 Rom. Rep. Phys. 59 759
[25] Zhang Y 2013 Acta Phys. Sin. 62 164501 (in Chinese) [张毅 2013 物理学报 62 164501]
[26] Chen J, Zhang Y 2014 Acta Phys. Sin. 63 104501 (in Chinese) [陈菊, 张毅 2014 物理学报 63 104501]
[27] Chen J, Zhang Y 2014 Nonlinear Dyn. 77 353
[28] Mei F X 1985 Foundations of Mechanics of Nonholonomic Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔1985非完整系统力学基础(北京: 北京工业学院出版社)]
[29] Mei F X, Wu H B 2009 Dynamics of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press)
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