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The problem of perturbation to Noether symmetry and adiabatic invariant for a nonconservative dynamic system is studied under a dynamic model presented by El-Nabulsi. First of all, the fractional action-like variational problem proposed by El-Nabulsi under the framework of the fractional calculus and based on the definition of the Riemann-Liouville fractional integral is introduced, and the Euler-Lagrange equations of the nonconservative system are given. Secondly, the definition and criterion of the Noether quasi-symmetric transformation are given, the relationship between the Noether symmetry and the invariant is established, and the exact invariant is obtained. Finally, the perturbation to the Noether symmetry of the system after the action of a small disturbance and corresponding adiabatic invariant are proposed and studied, the conditions for the existence of adiabatic invariant and the formulation are given. An example is given to illustrate the application of results.
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Keywords:
- nonconservative system /
- El-Nabulsi dynamic model /
- perturbation of Noether symmetry /
- adiabatic invariant
[1] Riewe F 1996 Phys. Rev. E 53 1890
[2] Riewe F 1997 Phys. Rev. E 55 3581
[3] Agrawal O P 2001 J. Appl. Mech. 68 339
[4] Klimek M 2001 Czech. J. Phys. 51 1348
[5] Klimek M 2002 Czech. J. Phys. 52 1247
[6] Baleanu D, Avkar T 2004 Nuovo Cimento B 119 73
[7] Rabei E M, Alhalholy T S, Taani A A 2004 Turk. J. Phys. 28 213
[8] Narakari Achar B N, Hanneken J W, Clarke T 2004 Physica A 339 311
[9] Baleanu D 2006 Czech. J. Phys. 56 1087
[10] Tarasov V E 2006 J. Phys. A 39 8409
[11] Baleanu D, Muslih S I 2008 J. Vib. Contr. 14 1301
[12] Baleanu D, Trujillo J J 2009 Phys. Scr. 80 055101
[13] Atanacković T M, Konjik S, Pilipović S 2008 J. Phys. A: Math. Theor. 41 095201
[14] Wang Z H, Hu H Y 2009 Sci. China G 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑 39 1495]
[15] El-Nabulsi R A 2011 Centr. Eur. J. Phys. 9 250
[16] Cresson J, Inizan P 2009 Phys. Scr. T136 014007
[17] Almeida R, Malinowska A B, Torres D F M 2010 J. Math. Phys. 51 033503
[18] Zhou S, Fu H, Fu J L 2011 Sci. China: Phys. Mech. Astron. 54 1847
[19] Golmankhaneh A K, Golmankhaneh A K, Baleanu D, Baleanu M C 2010 Int. J. Theor. Phys. 49 365
[20] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 110505]
[21] EI-Nabulsi A R 2005 Fizika A 14 289
[22] El-Nabulsi A R, Torres D F M 2008 J. Math. Phys. 49 053521
[23] El-Nabulsi A R 2009 Chaos Soliton. Fract. 42 52
[24] El-Nabulsi A R 2011 Centr. Eur. J. Phys. 9 250
[25] El-Nabulsi A R 2011 Appl. Math. Comput. 217 9492
[26] Frederico G S F, Torres D F M 2006 Int. J. Appl. Math. 19 97
[27] Frederico G S F, Torres D F M 2007 Int. J. Ecol. Econ. Stat. 9(F07) 74
[28] Zhang Y, Zhou Y 2013 Nonlinear Dyn. DOI: 10.1007/s11071-013-0831-x
[29] Mei F X, Liu D, Luo Y 1991 Advanced Analytical Mechanics (Beijing: Beijing Institute of Technology Press) p728 (in Chinese) [梅凤翔, 刘端, 罗勇 1991 高等分析力学 (北京: 北京理工大学出版社) 第728页]
[30] 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]
[31] 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]
[32] Chen X W, Shang M, Mei F X 2001 Chin. Phys. 10 997
[33] Zhang Y 2002 Acta Phys. Sin. 51 1666 (in Chinese) [张毅 2002 51 1666]
[34] Chen X W, Wang X M, Wang M Q 2004 Chin. Phys. 13 2003
[35] Chen X W, Li Y M 2005 Chin. Phys. 14 663
[36] Zhang Y, Fan C X, Mei F X 2006 Acta Phys. Sin. 55 3237 (in Chinese) [张毅, 范存新, 梅凤翔 2006 55 3237]
[37] Zhang Y 2006 Acta Phys. Sin. 55 3833 (in Chinese) [张毅 2006 55 3833]
[38] Luo S K, Guo Y X 2007 Commun. Theor. Phys. (Beijing, China) 47 25
[39] Zhang Y 2006 Chin. Phys. 15 1935
[40] Zhang Y 2007 Acta Phys. Sin. 56 1855 (in Chinese) [张毅 2007 56 1855]
[41] Zhang Y, Fan C X 2007 Commun. Theor. Phys. (Beijing, China) 47 607
[42] Ding N, Fang J H, Wang P, Zhang X N 2008 Commun. Theor. Phys. (Beijing, China) 49 57
[43] Zhang M J, Fang J H, Lu K, Pang T, Lin P 2009 Commun. Theor. Phys. (Beijing, China) 51 961
-
[1] Riewe F 1996 Phys. Rev. E 53 1890
[2] Riewe F 1997 Phys. Rev. E 55 3581
[3] Agrawal O P 2001 J. Appl. Mech. 68 339
[4] Klimek M 2001 Czech. J. Phys. 51 1348
[5] Klimek M 2002 Czech. J. Phys. 52 1247
[6] Baleanu D, Avkar T 2004 Nuovo Cimento B 119 73
[7] Rabei E M, Alhalholy T S, Taani A A 2004 Turk. J. Phys. 28 213
[8] Narakari Achar B N, Hanneken J W, Clarke T 2004 Physica A 339 311
[9] Baleanu D 2006 Czech. J. Phys. 56 1087
[10] Tarasov V E 2006 J. Phys. A 39 8409
[11] Baleanu D, Muslih S I 2008 J. Vib. Contr. 14 1301
[12] Baleanu D, Trujillo J J 2009 Phys. Scr. 80 055101
[13] Atanacković T M, Konjik S, Pilipović S 2008 J. Phys. A: Math. Theor. 41 095201
[14] Wang Z H, Hu H Y 2009 Sci. China G 39 1495 (in Chinese) [王在华, 胡海岩 2009 中国科学G辑 39 1495]
[15] El-Nabulsi R A 2011 Centr. Eur. J. Phys. 9 250
[16] Cresson J, Inizan P 2009 Phys. Scr. T136 014007
[17] Almeida R, Malinowska A B, Torres D F M 2010 J. Math. Phys. 51 033503
[18] Zhou S, Fu H, Fu J L 2011 Sci. China: Phys. Mech. Astron. 54 1847
[19] Golmankhaneh A K, Golmankhaneh A K, Baleanu D, Baleanu M C 2010 Int. J. Theor. Phys. 49 365
[20] Shen Y J, Yang S P, Xing H J 2012 Acta Phys. Sin. 61 110505 (in Chinese) [申永军, 杨绍普, 邢海军 2012 61 110505]
[21] EI-Nabulsi A R 2005 Fizika A 14 289
[22] El-Nabulsi A R, Torres D F M 2008 J. Math. Phys. 49 053521
[23] El-Nabulsi A R 2009 Chaos Soliton. Fract. 42 52
[24] El-Nabulsi A R 2011 Centr. Eur. J. Phys. 9 250
[25] El-Nabulsi A R 2011 Appl. Math. Comput. 217 9492
[26] Frederico G S F, Torres D F M 2006 Int. J. Appl. Math. 19 97
[27] Frederico G S F, Torres D F M 2007 Int. J. Ecol. Econ. Stat. 9(F07) 74
[28] Zhang Y, Zhou Y 2013 Nonlinear Dyn. DOI: 10.1007/s11071-013-0831-x
[29] Mei F X, Liu D, Luo Y 1991 Advanced Analytical Mechanics (Beijing: Beijing Institute of Technology Press) p728 (in Chinese) [梅凤翔, 刘端, 罗勇 1991 高等分析力学 (北京: 北京理工大学出版社) 第728页]
[30] 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]
[31] 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]
[32] Chen X W, Shang M, Mei F X 2001 Chin. Phys. 10 997
[33] Zhang Y 2002 Acta Phys. Sin. 51 1666 (in Chinese) [张毅 2002 51 1666]
[34] Chen X W, Wang X M, Wang M Q 2004 Chin. Phys. 13 2003
[35] Chen X W, Li Y M 2005 Chin. Phys. 14 663
[36] Zhang Y, Fan C X, Mei F X 2006 Acta Phys. Sin. 55 3237 (in Chinese) [张毅, 范存新, 梅凤翔 2006 55 3237]
[37] Zhang Y 2006 Acta Phys. Sin. 55 3833 (in Chinese) [张毅 2006 55 3833]
[38] Luo S K, Guo Y X 2007 Commun. Theor. Phys. (Beijing, China) 47 25
[39] Zhang Y 2006 Chin. Phys. 15 1935
[40] Zhang Y 2007 Acta Phys. Sin. 56 1855 (in Chinese) [张毅 2007 56 1855]
[41] Zhang Y, Fan C X 2007 Commun. Theor. Phys. (Beijing, China) 47 607
[42] Ding N, Fang J H, Wang P, Zhang X N 2008 Commun. Theor. Phys. (Beijing, China) 49 57
[43] Zhang M J, Fang J H, Lu K, Pang T, Lin P 2009 Commun. Theor. Phys. (Beijing, China) 51 961
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