广义Birkhoff方程的积分方法

李彦敏; 梅凤翔

doi:10.7498/aps.59.5930

摘要

场方法和最终乘子法是求解运动微分方程的基本方法. 本文将这两种方法应用于广义Birkhoff系统,求出了场方法的基本偏微分方程和该方程的完全积分; 根据Jacobi最终乘子定理求出了广义Birkhoff方程的解. 并举例说明结果的应用.

关键词:

The field method and the last multiplier method are general integral methods for solving the differential equations of motion. The two methods are applied to the generalized Birkhoff system,and the complete integrals of the basic partial differential equation are given. Furthermore,the solutions of the generalized Birkhoff equations are obtained by Jacobi last multiplier theorem. An example is given to illustrate the application of the results.

Keywords:

作者及机构信息

李彦敏²,
梅凤翔¹

(1)北京理工大学宇航学院力学系,北京 100081; (2)商丘师范学院物理与信息工程系,商丘 476000

基金项目: 国家自然科学基金(批准号:10772025,10932002,10972127)和河南省自然科学基金(批准号:082300410330,082300410370)资助的课题.

Authors and contacts

Li Yan-Min²,
Mei Feng-Xiang¹

(1)Department of Applied Mechanics,School of Aerospace,Beijing Institute of Technology,Beijing 100081,China; (2)Department of Physics and Information Engineering,Shangqiu Normal University,Shangqiu 476000,China

参考文献

[1]	Birkhoff G D 1927 Dynamical Systems (Providence: AMS College Publisher)
[2]	Santilli R M 1978 Foundations of theoretical mechanicsⅠ(New York: Springer Verlag)
[3]	Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York: Springer Verlag)
[4]	Mei F X,Shi R C,Zhang Y F and Wu H B 1996 Dynamics of Birkhoff Systems (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]	Zhang H B 2001 Acta Phys. Sin. 50 1837 (in Chinese) [张宏彬 2001 50 1837]
[6]	Guo Y X,Luo S K,Shang M 2001 Rep. Math. Phys. 47 313
[7]	Luo S K,Lu Y B,Zhou Q,Wang Y D,Ou Y S 2002 Acta Phys. Sin. 51 1913 (in Chinese) [罗绍凯、卢一兵、周强、王应德、欧阳实 2002 51 1913]
[8]	Shang M,Guo Y X,Mei F X 2007 Chin. Phys. 16 292
[9]	Ge W K,Mei F X 2007 Acta Phys. Sin. 56 2476 (in Chinese) [葛伟宽、梅凤翔 2007 56 2479]
[10]	Mei F X,Gang T Q,Xie J F 2006 Chin. Phys. 15 1678
[11]	Fu J L,Chen L Q,Luo S K,Chen X W,Wang X M 2001 Acta Phys. Sin. 50 2289 (in Chinese) [傅景礼、陈立群、罗绍凯、陈向炜、王新民 2001 50 2289]
[12]	Zhang Y 2008 Acta Phys. Sin. 57 5374 (in Chinese) [张毅 2008 57 5374]
[13]	Gu S L,Zhang H B 2004 Chin. Phys. 13 979
[14]	Ding N,Fang J H,Chen X X 2008 Chin. Phys. B 17 1967
[15]	Chen X W,Zhang R C,Mei F X 2000 Acta Mech. Sin. 16 282
[16]	Chen X W,Mei F X 2000 Mechanics Research Communications 27 365
[17]	Chen X W 2002 Chin. Phys. 11 441
[18]	Li Y M 2008 J. of Henan Normal University 36 52 (in Chinese) [李彦敏 2008 河南师范大学学报(自然科学版) 36 52]
[19]	Mei F X 1993 Science in China Serie A 36 1456
[20]	Mei F X,Zhang Y F,He G 2007 J. of Beijing Institute of Technology 27 1035 (in Chinese) [梅凤翔、张永发、何光 2007 北京理工大学学报 27 1035]
[21]	Mei F X,Xie J F,Gang T Q 2008 Acta Phys. Sin. 57 4649 (in Chinese) [梅凤翔、谢加芳、冮铁强 2008 57 4649]
[22]	Mei F X,Cai J L 2008 Acta Phys. Sin. 57 4657(in Chinese) [梅凤翔、蔡建乐 2008 57 4657]
[23]	Ge W K,Mei F X 2009 Acta Phys. Sin. 58 699 (in Chinese) [葛伟宽、梅凤翔 2009 58 699]
[24]	Mei F X,Xie J F,Gang T Q 2008 Acta Mech. Sin. 24 583
[25]	Vujanovic ＇ B 1984 Int. J. Non-Linear Mech. 19 383
[26]	Mei F X 1989 Acta Mech. Sin. 5 260
[27]	Mei F X 1990 Acta Mech. Sin. 6 160
[28]	Whittaker E T 1952 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge: Vniv Press)
[29]	Mei F X,Shang M 2008 Chin. Phys. Lett. 25 3837

施引文献

[1]	Birkhoff G D 1927 Dynamical Systems (Providence: AMS College Publisher)
[2]	Santilli R M 1978 Foundations of theoretical mechanicsⅠ(New York: Springer Verlag)
[3]	Santilli R M 1983 Foundations of Theoretical Mechanics Ⅱ (New York: Springer Verlag)
[4]	Mei F X,Shi R C,Zhang Y F and Wu H B 1996 Dynamics of Birkhoff Systems (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]	Zhang H B 2001 Acta Phys. Sin. 50 1837 (in Chinese) [张宏彬 2001 50 1837]
[6]	Guo Y X,Luo S K,Shang M 2001 Rep. Math. Phys. 47 313
[7]	Luo S K,Lu Y B,Zhou Q,Wang Y D,Ou Y S 2002 Acta Phys. Sin. 51 1913 (in Chinese) [罗绍凯、卢一兵、周强、王应德、欧阳实 2002 51 1913]
[8]	Shang M,Guo Y X,Mei F X 2007 Chin. Phys. 16 292
[9]	Ge W K,Mei F X 2007 Acta Phys. Sin. 56 2476 (in Chinese) [葛伟宽、梅凤翔 2007 56 2479]
[10]	Mei F X,Gang T Q,Xie J F 2006 Chin. Phys. 15 1678
[11]	Fu J L,Chen L Q,Luo S K,Chen X W,Wang X M 2001 Acta Phys. Sin. 50 2289 (in Chinese) [傅景礼、陈立群、罗绍凯、陈向炜、王新民 2001 50 2289]
[12]	Zhang Y 2008 Acta Phys. Sin. 57 5374 (in Chinese) [张毅 2008 57 5374]
[13]	Gu S L,Zhang H B 2004 Chin. Phys. 13 979
[14]	Ding N,Fang J H,Chen X X 2008 Chin. Phys. B 17 1967
[15]	Chen X W,Zhang R C,Mei F X 2000 Acta Mech. Sin. 16 282
[16]	Chen X W,Mei F X 2000 Mechanics Research Communications 27 365
[17]	Chen X W 2002 Chin. Phys. 11 441
[18]	Li Y M 2008 J. of Henan Normal University 36 52 (in Chinese) [李彦敏 2008 河南师范大学学报(自然科学版) 36 52]
[19]	Mei F X 1993 Science in China Serie A 36 1456
[20]	Mei F X,Zhang Y F,He G 2007 J. of Beijing Institute of Technology 27 1035 (in Chinese) [梅凤翔、张永发、何光 2007 北京理工大学学报 27 1035]
[21]	Mei F X,Xie J F,Gang T Q 2008 Acta Phys. Sin. 57 4649 (in Chinese) [梅凤翔、谢加芳、冮铁强 2008 57 4649]
[22]	Mei F X,Cai J L 2008 Acta Phys. Sin. 57 4657(in Chinese) [梅凤翔、蔡建乐 2008 57 4657]
[23]	Ge W K,Mei F X 2009 Acta Phys. Sin. 58 699 (in Chinese) [葛伟宽、梅凤翔 2009 58 699]
[24]	Mei F X,Xie J F,Gang T Q 2008 Acta Mech. Sin. 24 583
[25]	Vujanovic ＇ B 1984 Int. J. Non-Linear Mech. 19 383
[26]	Mei F X 1989 Acta Mech. Sin. 5 260
[27]	Mei F X 1990 Acta Mech. Sin. 6 160
[28]	Whittaker E T 1952 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge: Vniv Press)
[29]	Mei F X,Shang M 2008 Chin. Phys. Lett. 25 3837

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