Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

doi:10.1007/s10483-012-1631-7

Applied Mathematics and Mechanics (English Edition) ›› 2012, Vol. 33 ›› Issue (11): 1393-1402.doi: https://doi.org/10.1007/s10483-012-1631-7

Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

黄卫立^1,2, 蔡建乐²

1. Department of Physics and Telecom Engineering, Hunan City University, Yiyang 413000, Hunan Province, P. R. China;
2. College of Science, Hangzhou Normal University, Hangzhou 310018, P. R. China

收稿日期:2012-03-12 修回日期:2012-05-05 出版日期:2012-11-10 发布日期:2012-11-10
通讯作者: Jian-le CAI, Professor, E-mail: caijianle@yahoo.com.cn E-mail:caijianle@yahoo.com.cn
基金资助:
Project supported by the National Natural Science Foundation of China (No. 10932002) and the Natural Science Foundation of Zhejiang Province of China (No. LY12A02008)

Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

Wei-li HUANG^1,2, Jian-le CAI²

1. Department of Physics and Telecom Engineering, Hunan City University, Yiyang 413000, Hunan Province, P. R. China;
2. College of Science, Hangzhou Normal University, Hangzhou 310018, P. R. China

Received:2012-03-12 Revised:2012-05-05 Online:2012-11-10 Published:2012-11-10
Supported by:
Project supported by the National Natural Science Foundation of China (No. 10932002) and the Natural Science Foundation of Zhejiang Province of China (No. LY12A02008)

摘要/Abstract

摘要： Conformal invariance and conserved quantities for a nonholonomic system of Chetaev’s type with variable mass are studied. The conformal factor expressions are derived. The necessary and sufficient conditions are obtained to make the system’s conformal invariance Lie symmetrical. The conformal invariance of the weak and strong Lie symmetries for the system is given. The corresponding conserved quantities of the system are derived. Finally, an application of the result is shown with an example.

关键词: nonholonomic system, variable mass, conformal invariance, conserved quantity, polar continua, dynamical equations, anholonomic physical frame

Abstract: Conformal invariance and conserved quantities for a nonholonomic system of Chetaev’s type with variable mass are studied. The conformal factor expressions are derived. The necessary and sufficient conditions are obtained to make the system’s conformal invariance Lie symmetrical. The conformal invariance of the weak and strong Lie symmetries for the system is given. The corresponding conserved quantities of the system are derived. Finally, an application of the result is shown with an example.

Key words: nonholonomic system, variable mass, conformal invariance, conserved quantity, polar continua, dynamical equations, anholonomic physical frame

中图分类号:

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.

Wei-li HUANG;Jian-le CAI. Conformal invariance for nonholonomic system of Chetaev’s type with variable mass[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(11): 1393-1402.

参考文献

[1] Yang, L. W. and Mei, F. X. Mechanics Variable Mass System, Beijing Institute of TechnologyPress, Beijing (1989)
[2] Mei, F. X. Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems,Science Press, Beijing (1999)
[3] Zhao, Y. Y. and Mei, F. X. Symmetries and Invariants of Mechanical Systems, Science Press,Beijing (1999)
[4] Mei, F. X. Symmetries and Conserved Quantities of Constrained Mechanical Systems, BeijingInstitute of Technology Press, Beijing (2004)
[5] Luo, S. K. and Zhang, Y. F. Advances in the Study of Dynamics of Constrained Systems, SciencePress, Beijing (2008)
[6] Fu, J. L., Chen, B. Y., and Chen, L. Q. Noether symmetries of discrete nonholonomic dynamicalsystems. Physics Letters A, 373, 409-412 (2009)
[7] Wang, P., Fang, J. H., and Wang, X. M. Discussion on perturbation to weak Noether symmetryand adiabatic invariants for Lagrange systems. Chinese Physics Letters, 26, 034501 (2009)
[8] Fu, J. L., Chen, L. Q., and Chen, B. Y. Noether-type theory for discrete mechanico-electricaldynamical systems with nonregular lattices. Science China: Physics, Mechanics & Astronomy,53, 1687-1698 (2010)
[9] Xia, L. L. and Shan, L. F. Weak Noether symmetry for a nonholonomic controllable mechanicalsystem. Chinese Physics B, 19, 090302 (2010)
[10] Wang, P. Perturbation to Noether symmetry and Noether adiabatic invariants of discretemechanico-electrical systems. Chinese Physics Letters, 28, 040203 (2011)
[11] Mei, F. X. Lie symmetries and conserved quantities of Birkhoffian systems. Chinese Science Bulletin,44, 318-320 (1999)
[12] Luo, S. K. A set of Lie symmetrical conservation law for rotational relativistic Hamiltonian systems.Communication in Theoretical Physics, 40, 265-268 (2003)
[13] Fu, J. L. and Chen, B. Y. Hojman conserved quantities and Lie symmetries of non-conservativesystems. Modern Physics Letters B, 23, 1315-1322 (2009)
[14] Zhou, S., Fu, H., and Fu, J. L. Symmetry theories of Hamiltonian systems with fractional derivatives.Science China: Physics, Mechanics & Astronomy, 54, 1847-1853 (2011)
[15] Jiang, W. A., Li, L., Li, Z. J., and Luo, S. K. Lie symmetrical perturbation and a new type of non-Noether adiabatic invariants for disturbed generalized Birkhoffian systems. Nonlinear Dynamics,67, 1075-1081 (2012)
[16] Mei, F. X. Form invariance of Lagrange system. Journal of Beijing Institute of Technology, 9,120-124 (2000)
[17] Li, H. and Fang, J. H. Lie symmetry and Mei symmetry of a rotational relativistic system in phasespace. Chinese Physics, 13, 1187-1190 (2004)
[18] Jia, L. Q., Zheng, S. W., and Zhang, Y. Y. Mei symmetry and Mei conserved quantity of nonholonomicsystems of non-Chetaev’s type in event space. Acta Physica Sinica, 56, 5575-5579(2007)
[19] Xia, L. L. and Zhao, X. L. Generalized Mei conserved quantity of Mei symmetry for mechanicoelectricalsystems with nonholonomic controllable constraints. Chinese Physics Letters, 26, 010203(2009)
[20] Jia, L. Q., Xie, Y. L., and Luo, S. K. Mei conserved quantity deduced from Mei symmetry ofAppell equation in a dynamical system of relative motion. Acta Physica Sinica, 60, 040201 (2011)
[21] Xu, X. J., Qin, M. C., and Mei, F. X. Unified symmetry of holonomic mechanical systems. ChinesePhysics, 14, 1287-1289 (2005)
[22] Wu, H. B. and Mei, F. X. Symmetry of Lagrangians of holonomic systems in terms of quasicoordinates.Chinese Physics B, 18, 3145-3149 (2009)
[23] Jiang, W. A. and Luo, S. K. A new type of non-Noether exact invariants and adiabatic invariantsof generalized Hamiltonian systems. Nonlinear Dynamics, 67, 475-482 (2012)
[24] Xia, L. L., Li, Y. C., Hou, Q. B., and Wang, J. Unified symmetry of nonholonomic mechanicalsystems with variable mass. Chinese Physics, 15, 903-906 (2004)
[25] Cui, J. C., Zhang, Y. Y., Yang, X. F., and Jia, L. Q. Mei symmetry and Mei conserved quantityof Appell equations for avariable mass holonomic system. Chinese Physics B, 19, 030304 (2010)
[26] Zhang, M. L., Sun, X. T., Wang, X. X., Xie, Y. L., and Jia, L. Q. Lie symmetry and the generalizedHojman conserved quantity of Nielsen equations for a variable mass holonomic system of relativemotion. Chinese Physics B, 20, 110202 (2011)
[27] Wu, H. B. and Mei, F. X. Symmetry of Lagrangians of holonomic variable mass system. ChinesePhysics B, 21, 064501 (2012)
[28] Galiullin, A. S., Gafarov, G. G., Malaishka, R. P., and Khwan, A. M. Analytical Dynamics ofHelmholtz, Birkhoff and Nambu Systems, RZUFN, Moscow (1997)
[29] Cai, J. L. and Mei, F. X. Conformal invariance and conserved quantity of Lagrange systems underLie point transformation. Acta Physica Sinica, 57, 5369-5373 (2008)
[30] Cai, J. L. Conformal invariance and conserved quantities of general holonomic systems. ChinesePhysics Letters, 25, 1523-1526 (2008)
[31] Cai, J. L., Luo, S. K., and Mei, F. X. Conformal invariance and conserved quantity of Hamiltonsystems. Chinese Physics B, 17, 3170-3174 (2008)
[32] Cai, J. L. Conformal invariance and conserved quantities of Mei symmetry for Lagrange systems.Acta Physica Polonica A, 115, 854-856 (2009)
[33] Cai, J. L. Conformal invariance and conserved quantities of Mei symmetry for general holonomicsystems. Acta Physica Sinica, 58, 22-27 (2009)
[34] Cai, J. L. Conformal invariance and conserved quantity for the nonholonomic system of Chetaev’stype. International Journal of Theoretical Physics, 49, 201-211 (2010)
[35] He, G. and Mei, F. X. Conformal invariance and integration of first-order differential equations.Chinese Physics B, 17, 2764-2766 (2008)
[36] Luo, Y. P. Generalized conformal symmetries and its application of Hamilton systems. InternationalJournal of Theoretical Physics, 48, 2665-2671 (2009)
[37] Luo, Y. P. and Fu, J. L. Conformal invariance and Hojman conserved quantities for holonomicsystems with quasi-coordinates. Chinese Physics B, 19, 090303 (2010)
[38] Zhang, Y. Conformal invariance and Noether symmetry, Lie symmetry of holonomic mechanicalsystems in event space. Chinese Physics B, 18, 4636-4642 (2009)
[39] Zhang, M. J., Fang, J. H., and Lu, K. Conformal invariance and conserved quantity of third-orderLagrange equations for non-conserved mechanical systems. Chinese Physics B, 18, 4650-4656(2009)
[40] Huang, W. L. and Cai, J. L. Conformal invariance and conserved quantity of Mei symmetry forhigher-order nonholonomic system. Acta Mechanica, 223(2), 433-440 (2012)
[41] Cai, J. L. Conformal invariance of Mei symmetry for the holonomic system with variable mass.Chinese Journal of Physics, 48, 728-735 (2010)

Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

Conformal invariance for nonholonomic system of Chetaev’s type with variable mass

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Yong WANG, Chang LIU, Jing XIAO, Fengxiang MEI. Quasi-momentum theorem in Riemann-Cartan space[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(5): 733-746.
[2]	吴惠彬;梅凤翔. Form invariance and conserved quantity for weakly nonholonomic system[J]. Applied Mathematics and Mechanics (English Edition), 2014, 35(10): 1293-1300.
[3]	V.COVIC M.VESKOVIC D.DJURIC A.OBRADOVIC. On the stability of equilibria of nonholonomic systems with nonlinear constraints[J]. Applied Mathematics and Mechanics (English Edition), 2010, 31(6): 751-760.
[4]	慕小武;虞继敏;毕卫萍;程代展. ROBUST ADAPTIVE CONTROL OF NONHOLONOMIC SYSTEMS WITH UNCERTAINTIES[J]. Applied Mathematics and Mechanics (English Edition), 2004, 25(3): 329-337.
[5]	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.
[6]	王向东;徐小增;梁(汲金)廷. MAXIMUM PRINCIPLES FOR GENERALIZED SOLUTIONS OF QUASI-LINEAR ELLIPTIC EQUATIONS[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(4): 458-467.
[7]	戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅶ)—INCREMENTAL RATE TYPE[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(12): 1375-1380.
[8]	戴天民. RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅵ)—CONSERVATION LAWS OF MASS AND INERTIA[J]. Applied Mathematics and Mechanics (English Edition), 2003, 24(12): 1369-1374.
[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]	戴天民. RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA(Ⅰ)──MICROPOLAR THERMOELASTICITY[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(2): 119-126.
[11]	方建会;赵嵩卿;焦志勇. THE LIE SYMMETRIES AND CONSERVED QUANTITIES OF VARIABLE-MASS NONHOLONOMIC SYSTEM OF NON-CHETAEV'S TYPE IN PHASE SPACE[J]. Applied Mathematics and Mechanics (English Edition), 2002, 23(10): 1215-1220.
[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]	方建会. THE CONSERVATION LAW OF SECOND-ORDER NON-HOLONOMIC SYSTEM OF NON-CHETAEV’S TYPE[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(7): 836-840.
[14]	李元成;张毅;梁景辉. NOETHER'S THEOREM FOR NONHOLONOMIC SYSTEMS OF NON-CHETAEV'S TYPE WITH UNILATERAL CONSTRAINTS IN EVENT SPACE[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(5): 543-548.
[15]	傅景礼;陈向炜;罗绍凯. LIE SYMMETRIES AND CONSERVED QUANTITIES OF ROTATIONAL RELATIVISTIC SYSTEMS[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(5): 549-556.