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We consider the perturbed system as the combination of unperturbed system and perturbed term according to the characteristic of the first order approximate conserved quantities, and we suggest a new method to obtain the first order approximate conserved quantities by three steps: first, we select a suitable method to obtain the conserved quantity I0 of unperturbed system, second, we calculate the influence of perturbed terms on conserved quantity I0, and finally we obtain the first order approximate conserved quantities of the system by using the characteristic of the first order approximate conserved quantities. An actual two-dimensional nonlinear dynamics perturbed system is studied in this paper, and four stable first order approximate conserved quantities are obtained by using this new method. The expressions of first order approximate solution of the system are also obtained by transforming coordinates and using the perturbation method, and four special cases are discussed in this paper.
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Keywords:
- nonlinear perturbed /
- second-ordinary dynamics systems /
- first order approximate conserved quantities /
- transformation of coordinates
[1] Leach P G L, Moyo S, Cotsakis S, Lemmer R L 2001 J. Nonlinear Math. Phys. 1 139
[2] Govinder K S, Heil T G, Uzer T 1998 Phys. Lett. A 240 127
[3] Unal G 2000 Phys. Lett. A 266 106
[4] Dolapci I T, Pakdemirli M 2004 Int. J. Non-linear Mech. 39 1603
[5] Kara A H, Mahomed F M, Qadir A 2008 Nonlinear Dyn. 51 183
[6] Grebenev V N, Oberlack M 2007 J. Nonlinear Math. Phys. 14 157
[7] Johnpillai A G, Kara A H, Mahomed F M 2009 J. Comput. Appl. Math. 223 508
[8] Lou Z M 2010 Acta Phys. Sin. 59 6764 (in Chinese) [楼智美 2010 59 6764]
[9] Lou Z M, Mei F X, Chen Z D 2012 Acta Phys. Sin. 61 110204 (in Chinese) [楼智美, 梅凤翔, 陈子栋 2012 61 110204]
[10] Zhang Z Y, Yong X L, Chen Y F 2009 Chin. Phys. B 19 2629
[11] Dong W S, Wang B X, Fang J H 2011 Chin. Phys. B 20 010204
[12] Chen R, Xu X J 2012 Chin. Phys. B 21 094510
[13] Fang J H 2010 Chin. Phys. B 19 040301
[14] Wang X X, Han Y L, Zhang M L, Jia L Q 2013 Chin. Phys. B 22 020201
[15] Xie Y L, Jia L Q, Luo S K 2011 Chin. Phys. B 20 010203
[16] Han Y L, Sun X T, Zhang Y Y, Jia L Q 2013 Acta Phys. Sin. 62 160201 (in Chinese) [韩月林, 孙现亭, 张耀宇, 贾利群 2013 62 160201]
[17] Haas F, Goedert J 1996 J. Phys. A: Math. Gen. 29 4083
[18] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 54 1969]
[19] Lou Z M 2005 Acta Phys. Sin. 54 1460 (in Chinese) [楼智美 2005 54 1460]
[20] Kaushal R S, Gupta S 2001 J. Phys. A: Math. Gen. 34 9879
[21] Kaushal R S, Parashar D, Gupta S 1997 Ann. Phys. 259 233
[22] Lou Z M 2007 Chin. Phys. 16 1182
[23] Lou Z M 2007 Acta Phys. Sin. 56 2475 (in Chinese) [楼智美 2007 56 2475]
[24] Ding G T 2013 Acta Phys. Sin. 62 064502 (in Chinese) [丁光涛 2013 62 064502]
[25] Ding G T 2013 Acta Phys. Sin. 62 064501 (in Chinese) [丁光涛 2013 62 064501]
[26] Prelle M J, Singer M F 1983 Trans. Amer. Math. Soc. 279 215
[27] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Phys. A: Math. Gen. 39 L69
[28] Lou Z M 2010 Acta Phys. Sin. 59 719 (in Chinese) [楼智美 2010 59 719]
[29] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 110201 (in Chinese) [楼智美, 梅凤翔 2012 61 110201]
[30] Goldstein H (translated by Chen W X) 1986 Classical Mechanics (2nd Ed.) (Beijing: Science Press) pp627–629 (in Chinese) [戈德斯坦H著 (陈为恂译) 1986 经典力学 (第二版) (北京: 科学出版社) 第627–629页]
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[1] Leach P G L, Moyo S, Cotsakis S, Lemmer R L 2001 J. Nonlinear Math. Phys. 1 139
[2] Govinder K S, Heil T G, Uzer T 1998 Phys. Lett. A 240 127
[3] Unal G 2000 Phys. Lett. A 266 106
[4] Dolapci I T, Pakdemirli M 2004 Int. J. Non-linear Mech. 39 1603
[5] Kara A H, Mahomed F M, Qadir A 2008 Nonlinear Dyn. 51 183
[6] Grebenev V N, Oberlack M 2007 J. Nonlinear Math. Phys. 14 157
[7] Johnpillai A G, Kara A H, Mahomed F M 2009 J. Comput. Appl. Math. 223 508
[8] Lou Z M 2010 Acta Phys. Sin. 59 6764 (in Chinese) [楼智美 2010 59 6764]
[9] Lou Z M, Mei F X, Chen Z D 2012 Acta Phys. Sin. 61 110204 (in Chinese) [楼智美, 梅凤翔, 陈子栋 2012 61 110204]
[10] Zhang Z Y, Yong X L, Chen Y F 2009 Chin. Phys. B 19 2629
[11] Dong W S, Wang B X, Fang J H 2011 Chin. Phys. B 20 010204
[12] Chen R, Xu X J 2012 Chin. Phys. B 21 094510
[13] Fang J H 2010 Chin. Phys. B 19 040301
[14] Wang X X, Han Y L, Zhang M L, Jia L Q 2013 Chin. Phys. B 22 020201
[15] Xie Y L, Jia L Q, Luo S K 2011 Chin. Phys. B 20 010203
[16] Han Y L, Sun X T, Zhang Y Y, Jia L Q 2013 Acta Phys. Sin. 62 160201 (in Chinese) [韩月林, 孙现亭, 张耀宇, 贾利群 2013 62 160201]
[17] Haas F, Goedert J 1996 J. Phys. A: Math. Gen. 29 4083
[18] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 54 1969]
[19] Lou Z M 2005 Acta Phys. Sin. 54 1460 (in Chinese) [楼智美 2005 54 1460]
[20] Kaushal R S, Gupta S 2001 J. Phys. A: Math. Gen. 34 9879
[21] Kaushal R S, Parashar D, Gupta S 1997 Ann. Phys. 259 233
[22] Lou Z M 2007 Chin. Phys. 16 1182
[23] Lou Z M 2007 Acta Phys. Sin. 56 2475 (in Chinese) [楼智美 2007 56 2475]
[24] Ding G T 2013 Acta Phys. Sin. 62 064502 (in Chinese) [丁光涛 2013 62 064502]
[25] Ding G T 2013 Acta Phys. Sin. 62 064501 (in Chinese) [丁光涛 2013 62 064501]
[26] Prelle M J, Singer M F 1983 Trans. Amer. Math. Soc. 279 215
[27] Chandrasekar V K, Senthilvelan M, Lakshmanan M 2006 J. Phys. A: Math. Gen. 39 L69
[28] Lou Z M 2010 Acta Phys. Sin. 59 719 (in Chinese) [楼智美 2010 59 719]
[29] Lou Z M, Mei F X 2012 Acta Phys. Sin. 61 110201 (in Chinese) [楼智美, 梅凤翔 2012 61 110201]
[30] Goldstein H (translated by Chen W X) 1986 Classical Mechanics (2nd Ed.) (Beijing: Science Press) pp627–629 (in Chinese) [戈德斯坦H著 (陈为恂译) 1986 经典力学 (第二版) (北京: 科学出版社) 第627–629页]
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