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We have investigated the internal relation among the first integral, Hojman's method and Birkhoff's equations. There is a special equivalent relationship between Birkhoffian and Birkhoffian functions constructed by Hojman's method, and from this we can derive a more general form of Hojman's method. Then, combining the special equivalent relationship and Birkhoff's equations, we can derive the cyclic integral in Birkhoffian sense. An example is given to illustrate the application of the results.
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Keywords:
- Birkhoffian systems /
- Hojman' /
- s method /
- General Hojman'
[1] Birkhoff G D 1927 Dynamical systems (Providence R I: AMS College Publ)
[2] Santilli R M 1983 Foundations of theoretical mechanics I!I (New York: Springer-Verlag)
[3] Hojman S, Urrutia L F 1981 J. Math. Phys. 22 1896
[4] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) (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)]
[5] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[6] Mei F X 2009 Inverse Problems of Dynamics (Beijing: National Defense Industry Press) (in Chinese) [梅凤翔 2009 动力学逆问题 (北京: 国防工业出版社)]
[7] Ding G T 2008 Acta. Phys. Sin. 57 7415 (in Chinese) [丁光涛 2008 57 7415]
[8] Guo Y X, Liu C, Liu S X, Chang P 2009 Sci. Chin. E 52 761
[9] Guo Y X, Liu S X, Liu C 2009 Phys. Lett. A 373 3915
[10] Shang M, Mei F X 2009 Chin. Phys. B 18 3155
[11] Zhang Y 2010 Chin. Phys. B 19 080301
[12] Chen X W 2002 Global Analysis of Birkhoffian System (Henan: Henan University Press) (in Chinese) [陈向炜 2002 Birkhoff系统的全局分析(河南大学出版社)]
[13] Mei F X 1993 Sci. Chin. A 36 1456
[14] Wu H B, Mei F X 2011 Chin. Phys. B 20 104501
[15] Luo S K, Cai J L 2003 Chin. Phys. 12 357
[16] Zheng S W, Jia L Q 2006 Acta. Phys. Sin. 55 5590 (in Chinese) [郑世旺, 贾利群 2006 55 5590]
[17] Ding G T 2010 Acta. Phys. Sin. 59 3643 (in Chinese) [丁光涛 2010 59 3643]
[18] Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin. Phys. 16 582
[19] Fu J L, Chen L Q, Xue Y 2003 Acta. Phys. Sin. 52 256 (in Chinese) [傅景礼, 陈立群, 薛纭 2003 52 256]
[20] Liu S X, Liu C, Guo Y X 2011 Chin. Phys. B 20 034501
[21] Liu S X, Liu C, Guo Y X 2011 Acta. Phys. Sin. 60 064501 (in Chinese) [刘世兴, 刘畅, 郭永新 2011 60 064501]
[22] Mei F X, Cui J C 2011 J. Beijing Ins. Tech. 20 285
[23] Li Y M, Mei F X 2010 Chin. Phys. B 19 080302
[24] Liu C, Liu S X, Guo Y X 2011 Sci. Chin.: Tech. Sci. 54 2100
[25] Cui J C, Mei F X 2010 J. Dyn. Control 8 297 (in Chinese) [崔金超, 梅凤翔 2010 动力学与控制学报 8 297]
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[1] Birkhoff G D 1927 Dynamical systems (Providence R I: AMS College Publ)
[2] Santilli R M 1983 Foundations of theoretical mechanics I!I (New York: Springer-Verlag)
[3] Hojman S, Urrutia L F 1981 J. Math. Phys. 22 1896
[4] Mei F X, Shi R C, Zhang Y F, Wu H B 1996 Dynamics of Birkhoffian System (Beijing: Beijing Institute of Technology Press) (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)]
[5] Guo Y X, Liu C, Liu S X 2010 Commun. Math. 18 21
[6] Mei F X 2009 Inverse Problems of Dynamics (Beijing: National Defense Industry Press) (in Chinese) [梅凤翔 2009 动力学逆问题 (北京: 国防工业出版社)]
[7] Ding G T 2008 Acta. Phys. Sin. 57 7415 (in Chinese) [丁光涛 2008 57 7415]
[8] Guo Y X, Liu C, Liu S X, Chang P 2009 Sci. Chin. E 52 761
[9] Guo Y X, Liu S X, Liu C 2009 Phys. Lett. A 373 3915
[10] Shang M, Mei F X 2009 Chin. Phys. B 18 3155
[11] Zhang Y 2010 Chin. Phys. B 19 080301
[12] Chen X W 2002 Global Analysis of Birkhoffian System (Henan: Henan University Press) (in Chinese) [陈向炜 2002 Birkhoff系统的全局分析(河南大学出版社)]
[13] Mei F X 1993 Sci. Chin. A 36 1456
[14] Wu H B, Mei F X 2011 Chin. Phys. B 20 104501
[15] Luo S K, Cai J L 2003 Chin. Phys. 12 357
[16] Zheng S W, Jia L Q 2006 Acta. Phys. Sin. 55 5590 (in Chinese) [郑世旺, 贾利群 2006 55 5590]
[17] Ding G T 2010 Acta. Phys. Sin. 59 3643 (in Chinese) [丁光涛 2010 59 3643]
[18] Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin. Phys. 16 582
[19] Fu J L, Chen L Q, Xue Y 2003 Acta. Phys. Sin. 52 256 (in Chinese) [傅景礼, 陈立群, 薛纭 2003 52 256]
[20] Liu S X, Liu C, Guo Y X 2011 Chin. Phys. B 20 034501
[21] Liu S X, Liu C, Guo Y X 2011 Acta. Phys. Sin. 60 064501 (in Chinese) [刘世兴, 刘畅, 郭永新 2011 60 064501]
[22] Mei F X, Cui J C 2011 J. Beijing Ins. Tech. 20 285
[23] Li Y M, Mei F X 2010 Chin. Phys. B 19 080302
[24] Liu C, Liu S X, Guo Y X 2011 Sci. Chin.: Tech. Sci. 54 2100
[25] Cui J C, Mei F X 2010 J. Dyn. Control 8 297 (in Chinese) [崔金超, 梅凤翔 2010 动力学与控制学报 8 297]
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