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## General Hojman's method for the construction of Birkhoffian representation

Cui Jin-Chao, Zhao Zhe, Guo Yong-Xin
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• #### 摘要

研究第一积分、Hojman方法及Birkhoff方程之间的内在联系. Hojman方法构造的Birkhoff函数(组)满足的一个特定关系, 对此关系加以分析得到更为一般的广义Hojman方法. 再将此关系与Birkhoff方程相结合, 导出Birkhoff系统Hojman意义下的循环积分. 举例说明结论的应用.

#### Abstract

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.

#### 作者及机构信息

###### 1. 北京理工大学宇航学院, 北京 100081; 2. 沈阳药科大学医疗器械学院, 沈阳 110016; 3. 辽宁大学物理学院, 沈阳 110036
• 基金项目: 国家自然科学基金(批准号:10932002,11172120,11202090,10972127)资助的课题.

#### Authors and contacts

###### 1. School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China; 2. School of Medical Devices, Shenyang Pharmaceutical University, Shenyang 110016, China; 3. College of Physics, Liaoning University, Shenyang 110036, China
• Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 10932002, 11172120, 11202090, 10972127).

#### 参考文献

 [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 物理学报 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 物理学报 55 5590] [17] Ding G T 2010 Acta. Phys. Sin. 59 3643 (in Chinese) [丁光涛 2010 物理学报 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 物理学报 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 物理学报 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]

#### 施引文献

•  [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 物理学报 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 物理学报 55 5590] [17] Ding G T 2010 Acta. Phys. Sin. 59 3643 (in Chinese) [丁光涛 2010 物理学报 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 物理学报 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 物理学报 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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##### 出版历程
• 收稿日期:  2012-12-03
• 修回日期:  2013-01-08
• 刊出日期:  2013-05-05

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