FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS

Applied Mathematics and Mechanics (English Edition) ›› 2003, Vol. 24 ›› Issue (4): 468-478.

FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS

罗绍凯^1,2,3

1. Institute of Mathematical Mechanics and Mathematical Physics, Changsha University, Changsha 410003, P. R. China;
2. Department of Applied Physics, Hunan University, Changsha 410082, P. R. China;
3. Science College, Southern Yangtze University, Wuxi, Jiangsu 214063, P. R. China

收稿日期:2002-02-28 修回日期:2003-01-10 出版日期:2003-04-18 发布日期:2003-04-18
基金资助:
the National Natural Science Foundation of China (19972010):the Natural Science Foundation of Henan Province (984053100, 998040080):the Scientific Research Foundation of the Education Bureau of Hunan Province (02C033)

FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS

LUO Shao-kai^1,2,3

1. Institute of Mathematical Mechanics and Mathematical Physics, Changsha University, Changsha 410003, P. R. China;
2. Department of Applied Physics, Hunan University, Changsha 410082, P. R. China;
3. Science College, Southern Yangtze University, Wuxi, Jiangsu 214063, P. R. China

Received:2002-02-28 Revised:2003-01-10 Online:2003-04-18 Published:2003-04-18
Supported by:
the National Natural Science Foundation of China (19972010);the Natural Science Foundation of Henan Province (984053100, 998040080);the Scientific Research Foundation of the Education Bureau of Hunan Province (02C033)

摘要/Abstract

摘要： A form invariance of the relativistic Birkhoffian system is studied, and the conserved quantities of the system are obtained. Under the infinitesimal transformation of groups, the definition and criteria of the form invariance of the system were given. In view of the invariance of relativistic Pfaff-Birkhoff-D'Alembert principle under the infinitesimal transformation of groups, the theory of Noether symmetries of the relativistic Birkhoffian system were constructed. The relation between the form invariance and the Noether symmetry is studied, and the results show that the form invariance can also lead to the Noether symmetrical conserved quantity of the relativistic Birkhoffian system under certain conditions.

Abstract: A form invariance of the relativistic Birkhoffian system is studied, and the conserved quantities of the system are obtained. Under the infinitesimal transformation of groups, the definition and criteria of the form invariance of the system were given. In view of the invariance of relativistic Pfaff-Birkhoff-D'Alembert principle under the infinitesimal transformation of groups, the theory of Noether symmetries of the relativistic Birkhoffian system were constructed. The relation between the form invariance and the Noether symmetry is studied, and the results show that the form invariance can also lead to the Noether symmetrical conserved quantity of the relativistic Birkhoffian system under certain conditions.

中图分类号:

罗绍凯. FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(4): 468-478.

LUO Shao-kai. FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(4): 468-478.

参考文献

[1] Birkhoff G D. Dynamical System[M]. New York:AMS College Publ, Providence, RI, 1927.
[2] Santilli R M. Foundations of Theoretical Mechanics Ⅱ [M]. New York:Springer-Verlag, 1983, 110-280.
[3] MEI Feng-xiang. Noether theory of Birkhoffian system[J]. Science in China, Series A, 1993, 36(12):1456-1547.
[4] MEI Feng-xiang. Stability of equilibrium for the autonomous Birkhoffian system [J]. Chinese Science Bulletin, 1993, 38(10):816-819.
[5] WU Hui-bin, MEI Feng-xiang. Transformation theory of generalized Birkhoffian system[J]. Chinese Science Bulletin, 1995, 40 (10):885-888. (in Chinese).
[6] MEI Feng-xiang, Levesque E I. Generalized canonical realization and Birkhoff's realization of Chaplygin's nonholonomic system[J]. Transactions of the CSME, 1995, 19 (2):59-73.
[7] MEI Feng-xiang. Poisson's theory of Birkhoffian system[J]. Chinese Science Bulletin, 1996, 41(8):641-645.
[8] MEI Feng-xiang. Stability of motion for a constrained Birkhoffian system in terms of independent variables[J]. Applied Mathematics and Mechanics (English Edition ), 1997, 18(1):55-60.
[9] MEI Feng-xiang. The progress of research on dynamics of Birkhoff's system[J]. Advances in Mechanics, 1997, 27(4):436-446. (in Chinese).
[10] GUO Ying-xiang, MEI Feng-xiang. Integrability for Pfaff constrained systems:A geometrial theory[J]. Acta Mechanica Sinica, 1998, 14(1):85-91.
[11] MEI Feng-xiang, ZHANG Yong-fa, SHANG Mei. Lie symmetries and conserved quantifies of Birkhoffian system[J]. Mechanics Research Communications, 1999, 26(1):7-12.
[12] CHEN Xiang-wei, LUO Shao-kai, MEI Feng-xiang. Bifurcation of equilibrium points for secondorder autonomous Birkhoffian system[J]. Acta Mechanica Solida Sinica, 2000, 21(3):251-255. (in Chinese).
[13] GUO Yong-xin, LUO Shao-kai, SHANG Mei, et al. Birkhoffian formulations of nonholonomic constrained systems[J]. Reports on Mathematical Physics, 2001, 47 (3):313-322.
[14] MEI Feng-xiang, SHI Rong-chang, ZHANG Yong-fa, et al. Dynamics of Birkhoffian System[M]. Beijing:Beijing Institute of Technology Press, 1996. (in Chinese).
[15] MEI Feng-xiang. Applications of Lie Groups and Lie Algebras on Constrained Mechanical Systems[M]. Beijing:Science Press, 1999. (in Chinese).
[16] LI Zi-ping. Classical and Quantum Constrained Systems and Their Symmetries [M]. Beijing:Beijing Polytechnica University Press, 1993. (in Chinese).
[17] LUO Shao-kai. On the theory of relativistic analytical mechanics[J]. Teaching Material Communication, 1987, (5):31-34. (in Chinese).
[18] LUO Shao-kai. Relativistic variational principles and equations of motion of high-order nonlinear nonholonomic system[A]. In:WANG Zhao-lin Ed. Proc ICDVC[C]. Beijing:Peking University Press, 1990, 645-652.
[19] LUO Shao-kai. Dynamical theory of relativistic nonlinear nonholonomic system[J]. Shanghai Journal of Mechanics, 1991, 12(1):67-70. (in Chinese).
[20] LUO Shao-kai. Relativistic Hamilton principles and Lagrange equations in the generalized event space[J]. College Physics, 1992, 11(10):14-16. (in Chinese).
[21] LUO Shao-kai. Relativistic generalized Volterra equations of variable mass high-order nonlinear nonholonomic system[J]. Acta Mathematica Scientia, 1992, 12(Supplementary issue):27-29. (in Chinese).
[22] LUO Shao-kai. Relativistic variation principles and equations of motion for. variable mass controllable mechanics system[J]. Applied Mathematics and Mechanics (English Edition ), 1996, 17(7):683-692.
[23] LI Yuan-cheng, FANG Jian-hui. Unified form of variable mass of universal D'Alembert principle with relativity[J]. College Physics, 1991, 13(6):27-29. (in Chinese).
[24] FANG Jian-hui, LI Yuan-cheng. The variation principle of relativistic mechanics of variable mass system in the velocity space[J]. Mechanics and Practice, 1994, 13(5):19-20. (in Chinese).
[25] LUO Shao-kai. Rotational relativistic mechanics and rotational relativistic analytical mechanics[J]. Journal of Beijing Irstitute of Technology, 1996, 16(S1):154-158. (in Chinese).
[26] LUO Shao-kai. The theory of rotational relativistic analytical mechanics[J]. Applied Mathematics and Mechanics (English Edition ), 1998, 19(1 ):45-58.
[27] FU Jing-li, CHEN Xiang-wei, LUO Shao-kai. Algebraic structures and Poisson integrals of relativistic dynamical equations for rotational systems[J]. Applied Mathematics and Mechanics (Enslish Edition), 1999, 20(11 ):1266-1274.
[28] FU Jing-li, CHEN Xiang-wei, LUO Shao-kai. Lie symmetries and conserved quantities of rotational relativistic systens[J]. Applied Mathematics and Mechanics (English Edition ), 2000, 21(5):549-556.
[29] LUO Shao-kai, GUO Yong-xin, CHEN Xiang-wei, et al. Integration theory of the dynamics of a rotational relativistic system[J]. Acta Physica Sinica, 2001, 50 (11 ):2053-2058. (in Chinese).
[30] QIAO Yong-fen, LI Ren-Jie, MENG Jun. Lindelof's equations of nonholonomic rotational relativistic systems[J]. Acta Physica Sinica, 2001, 50(9):1637-1642. (in Chinese).
[31] FANG Jian-hui, ZHAO Song-qing. Lie symmetries and conserved quantities of relativistic rotational variable mass system[J]. Acta Physica Sinica, 2001, 50(3):390-393. (in Chinese).
[32] FU Jing-li, WANG Xin-min. Li symmetries and conserved quantities of relativistic Birkhoff systems[J]. Acta Physica Sinica, 2000, 49(6):1023-1029. (in Chinese).
[33] FU Jing-li, CHEN Xiang-wei, LUO Shao-kai. The Noether's theory of relativistic Birkhoff systems[J]. Acta Mechanica Solida Sinica, 2001, 22 (3):263-267. (in Chinese).
[34] FU Jing-li, CHEN Li-qun, LUO Shao-kai, et al. On the research for the dynamics of relativistic Birkhoff systems[J]. Acta Physica Sinica, 2001, 50 (12):2289-2295. (in Chinese).
[35] LUO Shao-kai, FU Jing-li, CHEN Xiang-wei. Basic theory of relativictic Birkhoffian dynamics of rotational system[J]. Acta Physica Sinica, 2001, 50(3):383-389. (in Chinese).
[36] LUO Shao-kai, CHEN Xiang-wei, FU Jing-li. Birkhoff's equations and geometrical theory of rotational relativistic system[J]. Chinese Physics, 2001, 10(4):271-276.
[37] LUO Shao-kai, GUO Yong-xin, CHEN Xiang-wei, et al. A field method for solving the equations of motion of a rotational relativistic Birkhoffian system[J]. Acta Physica Sinica, 2001, 50(11):2049-2052. (in Chinese).
[38] Noether A E. Invariance variations problems[J]. Kgl Ges Wiss Nachr Gottingen Math-Phys, 1918, (2):235-257.
[39] Candottie E, Palmieri C, Vitale B. On the inversion of Noether's theory in classical dynamical system[J]. America Journal of Physics, 1972, 40 (5):424-429.
[40] Djukic Dj S, Vujanovic B. Noether's theory in classical nonconservative mechanics[J]. Acta Mechanica, 1975, 23(1):17-27.
[41] Vujanovic B. Conservation laws of dynamical system via D'Alernbert principle[J]. International Journal of Non-Linear Mechanics, 1978, 13(2):185-197.
[42] Vujanovic B. A study of conservation laws of dynamical systems by means of the differential variational principles of Jourdain and Gauss[J]. Acta Mechanica, 1986, 65(1):63-80.
[43] LI Zi-ping. Symmetrical transformation of constrained system[J]. Acta Physica Sinica, 1981, 30 (12):1699-1705. (in Chinese).
[44] LI Zi-ping. Canonical form's Noether theorem and its inverse theorem of nonconservative requar system[J]. Chinese Science Bulletin, 1992, 37(23):2204-2205. (in Chinese).
[45] Bahar L Y, Kwatny H G. Extension of Noether's theory to constrained nonconservative dynamical systems[J]. International Journal of Non-Linear Mechanics, 1987, 22(2):125-138.
[46] LIU Duan. Conserved laws of nonholonomic nonconservative dynamical system[J]. Acta Mechanica Sinica, 1989, 21(1 ):75-83. (in Chinese).
[47] LIU Duan. Noether's theorem and Noether's inverse theorem of nonholonomic nonconservative dynamical system[J]. Science in China, Series A, 1991, 31(4):419-429. (in Chinese).
[48] LUO Shao-kai. Generalized Noether's theorem of nonholonomic nonpotential system in noninertial reference frame[J]. Applied Mathematics and Mechanics (English Edition ), 1991, 12(9):927-934.
[49] LUO Shao-kai. Generalized Noether's theorem of variable mass higher-order nonholonomic mechanical system in noninertial reference frame[J]. Chinese Science Bulletin, 1991, 36 (22):1930-1932. (in Chinese).
[50] LUO Shao-kai. Generalized conserved laws of relativistic mechanics[J]. Journal of Xinyang Teachers College, 1991, 4(4):57-64. (in Chinese).
[51] LUO Shao-kai. On the invariant theory of nonholonomic system with constraints of non-Chetaev type[J]. Acta Mechanica Solida Sinica, 1993, 6(1):47-57.
[52] ZNAO Yue-yu, MEI Feng-xiang. Symmetries and Conserved Quantities of Mechanical System[M]. Beijing:Science Press, 1999, 1-72. (in Chinese).

FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS

FORM INVARIANCE AND NOETHER SYMMETRICAL CONSERVED QUANTITY OF RELATIVISTIC BIRKHOFFIAN SYSTEMS

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 14

编辑推荐

Metrics

本文评价

[1]	K. REGER R. A. VAN GORDER. Lane-Emden equations of second kind modelling thermal explosion in infinite cylinder and sphere[J]. Applied Mathematics and Mechanics (English Edition), 2013, 34(12): 1439-1452.
[2]	黄卫立;蔡建乐. Conformal invariance for nonholonomic system of Chetaev’s type with variable mass[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(11): 1393-1402.
[3]	武秀根;郑百林;贺鹏飞;刘曙光. Equilibrium equations for 3D critical buckling of helical springs[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(8): 1049-1058.
[4]	李纬华;罗恩;黄伟江. Unconventional Hamilton-type variational principles for nonlinear elastodynamics of orthogonal cable-net structures[J]. Applied Mathematics and Mechanics (English Edition), 2007, 28(7): 931-942 .
[5]	张荣业. INVARIANT FORM AND INTEGRAL INVARIANTS ON KÄHLER MANIFOLD[J]. Applied Mathematics and Mechanics (English Edition), 2006, 27(2): 269-278 .
[6]	张荣业. LAGRANGIAN MECHANICS ON KAHLER MANIFOLDS[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(10): 1363-1374 .
[7]	方建会;陈培胜;张军. FORM INVARIANCE AND LIE SYMMETRY OF VARIABLE MASS NONHOLONOMIC MECHANICAL SYSTEM[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(2): 204-209 .
[8]	郭永新;尚玫;罗绍凯. POINCARÉ-CARTAN INTEGRAL INVARIANTS OF BIRKHOFFIAN SYSTEMS[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(1): 68-72.
[9]	方建会. LIE SYMMETRIES AND CONSERVED QUANTITIES OF SECOND-ORDER NONHOLONOMIC MECHANICAL SYSTEM[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(9): 1105-1110.
[10]	KH.I.Khalil. THE DRAG EXERTED BY AN OBLATE ROTATING ATMOSPHERE ON AN ARTIFICIAL SATELLITE[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(9): 1016-1028.
[11]	胡辉. PRELIMINARY ANALYSIS OF SKATING MOTION[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(1): 114-118.
[12]	方建会. THE CONSERVATION LAW OF NONHOLONOMIC SYSTEM OF SECOND-ORDER NON-CHETAVE’S TYPE IN EVENT SPACE[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(1): 89-94.
[13]	梁立孚;陈卫东. THE COMPLEMENTARY PROPERTY OF LINDELÖF’S WORK AND CHAPLYGIN’S WORK[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(8): 937-946.
[14]	张毅;梅凤翔. NOETHER’S THEORY OF MECHANICAL SYSTEMS WITH UNILATERAL CONSTRAINTS[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(1): 59-66.