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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.
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Keywords:
- generalized Birkhoff system /
- field method /
- last multiplier method /
- integral
[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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[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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