Abstract
Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré-Cartan's type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré's type is found.
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References
CHEN Bin.Analytical Dynamics[M]. Beijing: Peking University Press, 1987. (in Chinese)
LIU Duan, LUO Yong, XIN Shen-yu. About the basic integral variants of holonomic nonconservative dynamical systems[J].Acta Mechanica Sinica, 1991,7(2): 175–185.
Marsden J E, Ratiu T S.Introduction to Mechanics and Symmetry [M]. New York: Springer Verlag, 1994.
GUO Yong-xin, SHANG Mei, Mei Feng-xiang Poincaré-Cartan integral invariants of nonconservative dynamical systems[J].Internat J Theoret Phys., 1999,38 (3): 1017–1027.
GUO Yong-xin, SHANG Mei, LUO Shao-kai,et al. Poincaré-Cartan integral, variants and invariants of nonholonomic constrained systems [J].Intetnat J Theoret Phys., 2001,40(6): 1197–1205.
LIU Cheng-qun, LUO Shi-yu. Integral invariant in nonconservative systems and its application in modern physics [J].Applied Mathematics and Mechanics (English Edition), 1985,6(10): 949–956.
Santilli R M.Foundation of Theoretical Mechanics II [M]. New York: Springer-Verlag, 1983.
MEI Feng-xiang.Dynamics of Birkhoffian Systems [M] Beijing: Press of Beijing Institute of Technology, 1996. (in Chinese)
MEI Feng-xiang Progress in research to dynamics of Birkhoffian systems [J].Progress in Mechanics, 1997,27(4): 436–446. (in Chinese)
MEI Feng-xiang. Lie symmetry and conservation law of Birkhoffian system [J].Chinese Science Bulletin, 1994,44(4): 318–320.
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Communicated by Lin Zong-chi
Foundation items: the National Natural Science Foundation of China (10175032); the Natural Science Foundation of Liaoning Province of China (002083); the Natural Science Foundation of Henan Province of China (998040080); the Science Research Foundation of Liaoning Educational Committee of China (990111004, 20021004)
Biography: GUO Yong-xin (1963-) Professor, Doctor E-mail: guoyongxin@hotmail.com
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Yong-xin, G., Mei, S. & Shao-kai, L. Poincaré-cartan integral invariants of Birkhoffian systems. Appl Math Mech 24, 68–72 (2003). https://doi.org/10.1007/BF02439379
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DOI: https://doi.org/10.1007/BF02439379