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The method of constructing Birkhoffian and Birkhoffian funcations of mechanical system is studied. Based on the conditions of self-adjointness and integrability conditions of linear partial differential equations, a method of undetermined tensor for constructing Birkhoffian and Birkhoffian funcations of mechanical systems is given. Two examples are given to illustrate the application of the results.
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Keywords:
- Birkhoffian mechanics /
- conditions of self-adjointness /
- Birkhoffian and Birkhoffian funcations /
- method of undetermined tensor
[1] Birkhoff G D 1927 Dynamical systems (Providence R I: AMS College Publ)
[2] Santilli R M 1983 Foundations of theoretical mechanics Ⅱ (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 2011 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] Mei F X 1993 Sci. Chin. A 36 1456
[9] Wu H B, Mei F X 2011 Chin. Phys. B 20 104501
[10] Luo S K, Cai J L 2003 Chin. Phys. 12 357
[11] Ge W K, Zhang Y 2011 Acta. Phys. Sin. 60 050202 (in Chinese) [葛伟宽, 张毅 2011 物理学报 60 050202]
[12] Ding G T 2010 Acta. Phys. Sin. 59 3643 (in Chinese) [丁光涛 2010 物理学报 59 3643]
[13] Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin. Phys. 16 582
[14] 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]
[15] Guo Y X, Liu C, Liu S X, Chang P 2009 Sci. Chin. E 52 761
[16] Guo Y X, Liu S X, Liu C 2009 Phys. Lett. A 373 3915
[17] Shang M, Mei F X 2009 Chin. Phys. B 18 3155
[18] Zhang Y 2010 Chin. Phys. B 19 080301
[19] 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)]
[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 [崔金超, 梅凤翔 2010 动力学与控制学报 8 297]
[26] Kamke E 1974 Differentialgleichungen Losungsmethoden und Losungen, Bd. II: Partielle differentialgleichungen (Chelsea Publishing Co.)
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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 Ⅱ (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 2011 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] Mei F X 1993 Sci. Chin. A 36 1456
[9] Wu H B, Mei F X 2011 Chin. Phys. B 20 104501
[10] Luo S K, Cai J L 2003 Chin. Phys. 12 357
[11] Ge W K, Zhang Y 2011 Acta. Phys. Sin. 60 050202 (in Chinese) [葛伟宽, 张毅 2011 物理学报 60 050202]
[12] Ding G T 2010 Acta. Phys. Sin. 59 3643 (in Chinese) [丁光涛 2010 物理学报 59 3643]
[13] Zhang H B, Chen L Q, Gu S L, Liu C Z 2007 Chin. Phys. 16 582
[14] 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]
[15] Guo Y X, Liu C, Liu S X, Chang P 2009 Sci. Chin. E 52 761
[16] Guo Y X, Liu S X, Liu C 2009 Phys. Lett. A 373 3915
[17] Shang M, Mei F X 2009 Chin. Phys. B 18 3155
[18] Zhang Y 2010 Chin. Phys. B 19 080301
[19] 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)]
[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 [崔金超, 梅凤翔 2010 动力学与控制学报 8 297]
[26] Kamke E 1974 Differentialgleichungen Losungsmethoden und Losungen, Bd. II: Partielle differentialgleichungen (Chelsea Publishing Co.)
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