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For a generalized Hamiltonian system, Mei conserved quantity derived by using Mei symmetry is studied. First, the definition,the criterion and the determining equations of Mei symmetry of generalized Hamiltonian system are given under infinitesimal transformations of group. Second, the conditions and the forms for existence of Mei conserved quantity are directly obtained by using the Mei symmetry of the system. Then, the theorem for existence of Mei conserved quantity of generalized Hamiltonian system with additional terms is given. Finally, a new three-dimensional generalized Hamiltonian system and the plane motion of the three vortices of three-body problem are studied by using the method presented in the paper.
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Keywords:
- generalized Hamiltonian system /
- Mei symmetry /
- Mei conserved quantity /
- three-body problem
[1] Li J B, Zhao X H, Liu Z R 1994 Theory and Application of Generalized Hamilton Systems (Beijing: Science Press)(in Chinese) [李继彬、 赵晓华、 刘正荣 1994 广义哈密尔顿系统理论及其应用(北京: 科学出版社)]
[2] Zhao X H 1994 Acta Math. Appl. Sin. 17 182(in Chinese)[赵晓华 1994 应用数学学报 17 182]
[3] Olver P J 1986 Applications of Lie Groups to Differential Equations (New York: Spring-Verlag)
[4] Marsden J E, Ratiu T S 1994 Introduction to Mechanics and Symmetry (New York: Spring-Verlag)
[5] Mei F X 2003 Acta Phys. Sin. 52 1048(in Chinese)[梅凤翔 2003 52 1048]
[6] Jia L Q, Zheng S W 2006 Acta Phys. Sin. 55 3829(in Chinese)[贾利群、 郑世旺 2006 55 3829]
[7] Wu H B 2004 J. Beijing Inst. Technol. 24 20(in Chinese)[吴惠彬 2004 北京理工大学学报 24 20]
[8] Zhang S Y, Deng Z C 2004 J. Compt. Mech. 21 571(in Chinese)[张素英、 邓子辰 2004 计算力学学报 21 571]
[9] Huang Z L 2005 Ph.D. Dissertation (Hangzhou: Zhejiang University)(in Chinese)[黄志龙 2005 博士学位论文 (杭州: 浙江大学)]
[10] Wang Y Z, Cheng D Z, Li C W 2002 Acta Autom. Sin. 28 745(in Chinese)[王玉振、 程代展、 李春文 2002 自动化学报 28 745]
[11] Cheng D Z, Xi Z R, Lu Q, Mei S W 2000 Sci. China E 30 341(in Chinese)[程代展、 席在荣、 卢 强、 梅生伟 2000 中国科学E 30 341]
[12] Liu C, Liu S X, Mei F X, Guo Y X 2008 Acta Phys. Sin. 57 6709(in Chinese)[刘 畅、 刘世兴、 梅凤翔、 郭永新 2008 57 6709]
[13] Noether A E 1918 Nachr. Akad. Wiss. Gottingen: Math. Phys. 2 235
[14] Li Z P 1981 Acta Phys. Sin. 30 1659(in Chinese)[李子平 1981 30 1659]
[15] Li Z P 1993 Classical and Quantal Dynamics of Constrained Systems and Their Symmetrical Properies (Beijing: Beijing Polytechnic University Press)(in Chinese)[李子平 1993 经典和量子约束系统及其对称性 (北京: 北京工业大学出版社)]
[16] Mei F X 1993 Sci. China A 36 1456
[17] Liu D 1990 Sci. China A 11 1189 (in Chinese)[刘 端 1990 中国科学 A 11 1189]
[18] Luo S K 1991 Chin. Sci. Bull. 36 1930
[19] Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press)(in Chinese)[赵跃宇、 梅凤翔 1999 力学系统的对称性与不变量(北京:科学出版社)]
[20] Mei F X 2004 Symmetry and Conserved Quantity of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量 (北京:北京理工大学出版社)]
[21] Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese) [罗绍凯、 张永发 2008 约束系统动力学研究进展 (北京: 科学出版社)]
[22] Lutzky M 1979 J. Phys. A 12 973
[23] Lutzky M 1979 J. Math. Phys. A 19 105
[24] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用 (北京: 科学出版社)]
[25] Mei F X 1998 Chin. Sci. Bull. 43 1937
[26] Zhang Y, Xue Y 2001 Acta Phys. Sin. 50 816 (in Chinese)[张 毅、 薛 纭 2001 50 816]
[27] Zhang H B, Chen L Q, Liu R W, Gu S L 2005 Acta Phys. Sin. 54 2489 (in Chinese) [张宏彬、 陈立群、 刘荣万、 顾书龙 2005 54 2489]
[28] Guo Y X, Jiang L Y, Yu Y 2001 Chin. Phys. 10 181
[29] Luo S K, Jia L Q 2003 Commun. Theor. Phys. 40 265
[30] Luo S K 2003 Chin. Phys. Lett. 20 597
[31] Fan J H 2010 Chin. Phys. B 19 040301
[32] Mei F X 2000 J. Beijing Inst. Technol. 9 120
[33] Luo S K 2003 Acta Phys. Sin. 52 2941 (in Chinese) [罗绍凯 2003 52 2941]
[34] Fan J H 2003 Commun. Theor. Phys. 40 269
[35] Chen X W, Luo S K, Mei F X 2002 Appl. Math. Mech. 23 47(in Chinese) [陈向炜、 罗绍凯、 梅凤翔 2002 应用数学与力 学 23 47] 〖36] Wang S Y, Mei F X 2001 Chin. Phys. 10 373
[36] Luo S K 2002 Chin. Phys. Lett. 19 449
[37] Luo S K 2002 Commun. Theor. Phys. 38 257
[38] Ge W K 2002 Acta Phys. Sin. 51 939 (in Chinese)[葛伟宽 2002 51 939]
[39] Ge W K, Zhang Y 2006 Acta Phys. Sin. 55 4985 (in Chinese) [葛伟宽、 张 毅 2006 55 4985]
[40] Fang J H, Liao Y P, Peng Y 2005 Acta Phys. Sin. 54 496 (in Chinese) [方建会、 廖永潘、 彭 勇 2005 54 496]
[41] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 54 1969]
[42] Xia L L, Li Y C, Wang J, Hou Q B 2006 Commun. Theor. Phys. 46 415
[43] Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[44] Ding N, Fang J H 2008 Chin. Phys. B 17 1550
[45] Wang P, Fang J H, Wang X M 2009 Chin. Phys. B 18 1312
[46] Cui J C, Zhang Y Y, Jia L Q 2010 Chin. Phys. B 19 030304
[47] Qian M, Jiang Y P 1984 Acta Math. Sci. 3 441(in Chinese) [钱 敏、 蒋云平 1984 数学 3 441]
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[1] Li J B, Zhao X H, Liu Z R 1994 Theory and Application of Generalized Hamilton Systems (Beijing: Science Press)(in Chinese) [李继彬、 赵晓华、 刘正荣 1994 广义哈密尔顿系统理论及其应用(北京: 科学出版社)]
[2] Zhao X H 1994 Acta Math. Appl. Sin. 17 182(in Chinese)[赵晓华 1994 应用数学学报 17 182]
[3] Olver P J 1986 Applications of Lie Groups to Differential Equations (New York: Spring-Verlag)
[4] Marsden J E, Ratiu T S 1994 Introduction to Mechanics and Symmetry (New York: Spring-Verlag)
[5] Mei F X 2003 Acta Phys. Sin. 52 1048(in Chinese)[梅凤翔 2003 52 1048]
[6] Jia L Q, Zheng S W 2006 Acta Phys. Sin. 55 3829(in Chinese)[贾利群、 郑世旺 2006 55 3829]
[7] Wu H B 2004 J. Beijing Inst. Technol. 24 20(in Chinese)[吴惠彬 2004 北京理工大学学报 24 20]
[8] Zhang S Y, Deng Z C 2004 J. Compt. Mech. 21 571(in Chinese)[张素英、 邓子辰 2004 计算力学学报 21 571]
[9] Huang Z L 2005 Ph.D. Dissertation (Hangzhou: Zhejiang University)(in Chinese)[黄志龙 2005 博士学位论文 (杭州: 浙江大学)]
[10] Wang Y Z, Cheng D Z, Li C W 2002 Acta Autom. Sin. 28 745(in Chinese)[王玉振、 程代展、 李春文 2002 自动化学报 28 745]
[11] Cheng D Z, Xi Z R, Lu Q, Mei S W 2000 Sci. China E 30 341(in Chinese)[程代展、 席在荣、 卢 强、 梅生伟 2000 中国科学E 30 341]
[12] Liu C, Liu S X, Mei F X, Guo Y X 2008 Acta Phys. Sin. 57 6709(in Chinese)[刘 畅、 刘世兴、 梅凤翔、 郭永新 2008 57 6709]
[13] Noether A E 1918 Nachr. Akad. Wiss. Gottingen: Math. Phys. 2 235
[14] Li Z P 1981 Acta Phys. Sin. 30 1659(in Chinese)[李子平 1981 30 1659]
[15] Li Z P 1993 Classical and Quantal Dynamics of Constrained Systems and Their Symmetrical Properies (Beijing: Beijing Polytechnic University Press)(in Chinese)[李子平 1993 经典和量子约束系统及其对称性 (北京: 北京工业大学出版社)]
[16] Mei F X 1993 Sci. China A 36 1456
[17] Liu D 1990 Sci. China A 11 1189 (in Chinese)[刘 端 1990 中国科学 A 11 1189]
[18] Luo S K 1991 Chin. Sci. Bull. 36 1930
[19] Zhao Y Y, Mei F X 1999 Symmetries and Invariants of Mechanical Systems (Beijing: Science Press)(in Chinese)[赵跃宇、 梅凤翔 1999 力学系统的对称性与不变量(北京:科学出版社)]
[20] Mei F X 2004 Symmetry and Conserved Quantity of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese) [梅凤翔 2004 约束力学系统的对称性与守恒量 (北京:北京理工大学出版社)]
[21] Luo S K, Zhang Y F 2008 Advances in the Study of Dynamics of Constrained Systems (Beijing: Science Press) (in Chinese) [罗绍凯、 张永发 2008 约束系统动力学研究进展 (北京: 科学出版社)]
[22] Lutzky M 1979 J. Phys. A 12 973
[23] Lutzky M 1979 J. Math. Phys. A 19 105
[24] Mei F X 1999 Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems (Beijing: Science Press) (in Chinese) [梅凤翔 1999 李群和李代数对约束力学系统的应用 (北京: 科学出版社)]
[25] Mei F X 1998 Chin. Sci. Bull. 43 1937
[26] Zhang Y, Xue Y 2001 Acta Phys. Sin. 50 816 (in Chinese)[张 毅、 薛 纭 2001 50 816]
[27] Zhang H B, Chen L Q, Liu R W, Gu S L 2005 Acta Phys. Sin. 54 2489 (in Chinese) [张宏彬、 陈立群、 刘荣万、 顾书龙 2005 54 2489]
[28] Guo Y X, Jiang L Y, Yu Y 2001 Chin. Phys. 10 181
[29] Luo S K, Jia L Q 2003 Commun. Theor. Phys. 40 265
[30] Luo S K 2003 Chin. Phys. Lett. 20 597
[31] Fan J H 2010 Chin. Phys. B 19 040301
[32] Mei F X 2000 J. Beijing Inst. Technol. 9 120
[33] Luo S K 2003 Acta Phys. Sin. 52 2941 (in Chinese) [罗绍凯 2003 52 2941]
[34] Fan J H 2003 Commun. Theor. Phys. 40 269
[35] Chen X W, Luo S K, Mei F X 2002 Appl. Math. Mech. 23 47(in Chinese) [陈向炜、 罗绍凯、 梅凤翔 2002 应用数学与力 学 23 47] 〖36] Wang S Y, Mei F X 2001 Chin. Phys. 10 373
[36] Luo S K 2002 Chin. Phys. Lett. 19 449
[37] Luo S K 2002 Commun. Theor. Phys. 38 257
[38] Ge W K 2002 Acta Phys. Sin. 51 939 (in Chinese)[葛伟宽 2002 51 939]
[39] Ge W K, Zhang Y 2006 Acta Phys. Sin. 55 4985 (in Chinese) [葛伟宽、 张 毅 2006 55 4985]
[40] Fang J H, Liao Y P, Peng Y 2005 Acta Phys. Sin. 54 496 (in Chinese) [方建会、 廖永潘、 彭 勇 2005 54 496]
[41] Lou Z M 2005 Acta Phys. Sin. 54 1969 (in Chinese) [楼智美 2005 54 1969]
[42] Xia L L, Li Y C, Wang J, Hou Q B 2006 Commun. Theor. Phys. 46 415
[43] Cai J L 2009 Acta Phys. Sin. 58 22 (in Chinese) [蔡建乐 2009 58 22]
[44] Ding N, Fang J H 2008 Chin. Phys. B 17 1550
[45] Wang P, Fang J H, Wang X M 2009 Chin. Phys. B 18 1312
[46] Cui J C, Zhang Y Y, Jia L Q 2010 Chin. Phys. B 19 030304
[47] Qian M, Jiang Y P 1984 Acta Math. Sci. 3 441(in Chinese) [钱 敏、 蒋云平 1984 数学 3 441]
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