摘要We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.
Abstract：We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.
LI Yan-Min. Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System[J]. 中国物理快报, 2010, 27(1): 10202-010202.
LI Yan-Min. Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System. Chin. Phys. Lett., 2010, 27(1): 10202-010202.
[1] Birkhoff G D 1927 Dynamical Systems (Providence: AMSCollege Publisher) [2] Santilli R M 1978 Foundations of TheoreticalMechanics I (New York: Springer) [3] Santilli R M 1983 Foundations of TheoreticalMechanics I$\!$I (New York: Springer) [4] Mei F X, Shi R C, Zhang Y F and Wu H B 1996 Dynamicsof Birkhoff Systems (Beijing: Beijing Institute of Technology) (inChinese) [5] Zhang H B 2001 Acta Phys. Sin. 50 1837 (inChinese) [6] Guo Y X, Luo S K and Shang M 2001 Rep. Math. Phys. 47 313 [7] Luo S K, Lu Y B, Zhou Q, Wang Y D and Ou Y S 2002 Acta Phys. Sin. 51 1913 (in Chinese) [8] Shang M, Guo Y X and Mei F X 2007 Chin. Phys. 16 292 [9] Ge W K and Mei F X 2007 Acta Phys. Sin. 562476 (in Chinese) [10] Mei F X, Gang T Q and Xie J F 2006 Chin. Phys. 15 1678 [11] Fu J L, Chen L Q, Luo S K, Chen X W and Wang X M 2001 Acta Phys. Sin. 50 2289 (in Chinese) [12] Zhang Y 2008 Acta Phys. Sin. 57 5374 (inChinese) [13]Gu S L and Zhang H B 2004 Chin. Phys. 13 979 [14]Ding N, Fang J H and Chen X X 2008 Chin. Phys. B 17 1967 [15]Chen X W, Zhang R C and Mei F X 2000 Acta Mech. Sin. 16 282 [16]Chen X W and Mei F X 2000 Mech. Res. Commun. 27 365 [17]Chen X W 2002 Chin. Phys. 11 441 [18]Li Y M 2008 J. Henan Normal University 36 52(in Chinese) [19] Mei F X 1993 Sci. Chin. A 36 1456 [20] Mei F X, Zhang Y F and He G 2007 J. BeijingInstitute of Technology 27 1035 (in Chinese) [21] Mei F X, Xie J F and Gang T Q 2008 Acta Phys. Sin. 57 4649 (in Chinese) [22] Mei F X and Cai J L 2008 Acta Phys. Sin. 574657 (in Chinese) [23] Ge W K and Mei F X 2009 Acta Phys. Sin. 58699 (in Chinese) [24] Mei F X, Xie J F and Gang T Q 2008 Acta Mech. Sin. 24 583 [25] Guo Y X 2001 Chin. Phys. 10 181 [26] Mei F X, Zhang Y F and Shang M 1999 Mech. Res.Commun. 26 7 [27] Luo S K 2003 Commun. Theor. Phys. 40 265 [28] Ibragimov N H 1999 Elementary Lie Group Analysis andOrdinary Differential Equations (London: Wiley) [29] Borner H G and Davidson W F 1978 Phys. Rev. Lett. 40 167 [30] Fuchs J C 1991 J. Math. Phys. 32 1703 [31] Aguirre M and Krause J 1988 J. Math. Phys. 299 [32] Ibragimov N H 1996 CRC Handbook of Lie GroupAnalysis of Differential Equations (New York: CRC) vol 3 [33] Kruskal M 1962 J. Math. Phys. 3 806 [34] Djukic D J S 1981 Int. J. Nonlinear Mech. 16489 [35] Fu J L and Chen L Q 2004 Phys. Lett. A 324 95 [36] Hu J H 1999 SIAM J. Appl. Math. 59 322 [37] Zhao Y Y and Mei F X 1996 Acta Mech. Sin. 28207 (in Chinese). [38] Chen X W, Li Y M and Zhao Y H 2005 Phys. Lett. A 337 274 [39] Ostrovsky V N and Prudov N V 1995 J. Phys. B 20 4435
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