约束Hamilton系统的Lie对称性及其在场论中的应用

周景润; 傅景礼

doi:10.21656/1000-0887.390218

约束Hamilton系统的Lie对称性及其在场论中的应用

doi: 10.21656/1000-0887.390218

周景润^1,2,
傅景礼^1,2

¹绍兴职业技术学院理科教研室, 浙江绍兴 312000;²浙江理工大学数学物理研究所, 杭州 310018

基金项目: 国家自然科学基金(11272287；11872335；11472247);浙江省科技创新团队（2013TD18）

详细信息

作者简介:
周景润(1988—)，男，硕士(E-mail: 869569521@qq.com);傅景礼(1955—)，男，教授，博士，博士生导师(通讯作者. E-mail: sqfujingli@163.com).

中图分类号: O302;O34;O369
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- 文章访问数: 1135
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- PDF下载量: 320
- 被引次数: 0
出版历程
- 收稿日期: 2018-08-07
- 修回日期: 2018-09-07
- 刊出日期: 2019-07-01

Lie Symmetry of Constrained Hamiltonian Systems and Its Application in Field Theory

ZHOU Jingrun^1,2,
FU Jingli^1,2

¹Science Teaching and Research Section, Shaoxing Vocational and Technical College,Shaoxing, Zhejiang 312000, P.R.China;²Institute of Mathematical Physics, Zhejiang SciTech University,Hangzhou 310018, P.R.China

Funds: The National Natural Science Foundation of China(11272287;11872335;11472247)

摘要

摘要: 研究了约束Hamilton系统的Lie对称性，得到了场论系统的守恒量.首先给出约束Hamilton系统的正则运动方程和固有约束方程；其次构建了约束Hamilton 系统的Lie对称性确定方程和结构方程；然后给出了约束Hamilton系统的Lie守恒定理和守恒量；最后研究了复标量场与Chern-Simons项耦合系统的Lie对称性和另外一个例子以说明此方法在场论中的应用.
- Lie对称性 /
- 约束Hamilton系统 /
- 场论 /
- 守恒量
Abstract: The Lie symmetry method was studied for constrained Hamiltonian systems, and the conservation laws of the field theory systems were obtained. Firstly, the generalized canonical equations for constrained Hamiltonian systems were derived. Secondly, the determining equations and structural equations about the Lie symmetry of the constrained Hamiltonian systems were deduced. Thirdly, the Lie theorems and the conserved quantities for constrained Hamiltonian systems were given. Finally, the Lie symmetry for the system of the complex scalar field coupled to the Chern-Simons term was discussed. Two examples in the field theory illustrate the validity of this method.
- Lie symmetry /
- constrained Hamiltonian system /
- field theory /
- conserved quantity

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参考文献(51)

[1]	XUE Y, ZHANG Y. Lie symmetries of constrained Hamiltonian systems with constraints of the second kind[J]. Acta Physica Sinica,2001,50(5): 816-819.
[2]	LUO S K. Mei symmetry, Noether symmetry and Lie symmetry of Hamiltonian canonical equations for a singular system[J]. Acta Physica Sinica,2004,53(1): 5-10.
[3]	张素英, 邓子辰. 用投影方法求耗散广义Hamilton约束系统的李群积分[J]. 应用数学和力学, 2004,〖STHZ〗 25(4): 385-390.(ZHANG Suying, DENG Zichen. Lie group integration for constrained generalized Hamiltonian system with dissipation by projection method[J]. Applied Mathematics and Mechanics,2004,〖STHZ〗 25(4): 385-390.(in Chinese))
[4]	DU M L, DELOS J B. Effect of closed classical orbits on quantum spectra: ionization of atoms in a magnetic field[J]. Physical Review Letters,1987,58(17): 1731-1733.
[5]	王永龙, 赵德玉. 约束Hamilton系统对称性及应用[M]. 山东: 山东人民出版社, 2012.(WANG Yonglong, ZHAO Deyu. The Symmetry of Constrained Hamilton System and Its Application [M]. Shandong: Shandong People’s Publishing House, 2012.(in Chinese))
[6]	TINKHAM M. Group Theory and Quantum Mechanics [M]. New York: McGraw-Hill, 1964.
[7]	BISWAS T. Symmetry breaking of gauge theories via internal space dynamics[J]. Journal of High Energy Physics,2002,2(1): 135-146.
[8]	LUTZKY M. Dynamical symmetries and conserved quantities[J]. Journal of Physics A: Mathematical and General,1979,12(7): 973-981.
[9]	韩广才, 张耀良. 一类广义力学系统的能量方程[J]. 哈尔滨工程大学学报, 2001,22(4): 69-71.(HAN Guangcai, ZHANG Yaoliang. Energy equation of generalized mechanical system[J]. Journal of Harbin Engineering University,2001,22(4): 69-71.(in Chinese))
[10]	LIU R. Noether’s theorem and its inverse of nonholonomic nonconservative dynamical systems[J]. Science in China(Series A),1991,34(4): 37-47.
[11]	BLUMAN G W, KUMEI S. Symmetries and Differential Equations [M]. Berlin: Springer, 1989.
[12]	MACCALLUM M. Differential Equations: Their Solution Using Symmetries [M]. Cambridge: Cambridge University Press, 1989.
[13]	OVSIANNIKOV L V. Group Analysis of Differential Equations [M]. New York: Academic Press, 1982.
[14]	GORRINGE V M, LEACH P G L. Lie point symmetries for systems of second order linear ordinary differential equations[J]. Quaestiones Mathematicae,1988,〖STHZ〗 11(1): 95-117.
[15]	CHEH J, OLVER P J, POHJANPELTO J. Algorithms for differential invariants of symmetry groups of differential equations[J]. Foundations of Computational Mathematics,2008,8: 501-532.
[16]	OLVER P J, POHJANPELTO J. Moving frames for Lie pseudo-groups[J]. Canadian Journal of Mathematics,2008,60: 1336-1386.
[17]	MEI J Q, WANG H Y. The generating set of the differential invariant algebra and Maurer-Cartan equations of a (2+1)-dimensional Burgers equation[J]. Journal of Systems Science and Complexity,2013,26(2): 281-290.
[18]	BEFFA G. Poisson geometry of differential invariants of curves in some nonsemisimple homogeneous spaces[J]. Proceedings of the American Mathematical Society,2006,134(3): 779-791.
[19]	OLVER P J, POHJANPELTO J. Differential invariant algebras of Lie pseudo-groups[J]. Advances in Mathematics,2009,222(5): 1746-1792.
[20]	LISLE G I, REID G J. Symmetry classification using noncommutative invariant differential operators[J]. Foundations of Computational Mathematics,2006,6(3): 353-386.
[21]	FU J L, CHEN B Y, CHEN L Q. Noether symmetries of discrete nonholonomic dynamical systems[J]. Physics Letters A,2009,373(4): 409-412.
[22]	傅景礼, 陈立群, 陈本永. 非规范格子离散非保守系统的Noether理论[J]. 中国科学(G辑), 2009,39(9): 1320-1329.(FU Jingli, CHEN Liqun, CHEN Benyong. Noether-type theorem for discrete nonconservative dynamical systems with nonregular lattices[J]. Scientia Sinica(Series G),2009,39(9): 1320-1329.(in Chinese))
[23]	傅景礼, 陈立群, 陈本永. 非规范格子离散机电耦合动力系统的Noether理论[J]. 中国科学(G辑), 2010,40(2): 133-145.(FU Jingli, CHEN Liqun, CHEN Benyong. Noether-type theorem for discrete electromechanical coupling systems with nonregular lattices[J]. Scientia Sinica(Series G), 2010,40(2): 133-145.(in Chinese))
[24]	赵跃宇. 非保守力学系统的Lie对称性和守恒量[J]. 力学学报, 1994,26(3): 380-384.(ZHAO Yueyu. Lie symmetries and conserved quantities of non-conservative mechanical system[J]. Acta Mechanica Sinica,1994,26(3): 380-384.(in Chinese))
[25]	梅凤翔. 李群和李代数对约束力学系统的应用[M]. 北京: 科学出版社, 1999.(MEI Fengxiang. Applications of Lie Groups and Lie Algebras to Constrained Mechanical Systems [M]. Beijing: Science Press, 1999.(in Chinese))
[26]	MEI F X. Lie symmetries and conserved quantities of constrained mechanical systems[J]. Acta Mechanica,2000,141: 135-148.
[27]	ZHANG H B. Lie symmetries and conserved quantities of non-holonomic mechanical systems with unilateral Vacco constraints[J]. Chinese Physics,2002,11(1): 1-4.
[28]	周景润, 傅景礼. 约束Hamilton系统的积分因子及其守恒量[J]. 力学季刊, 2018,39(3): 554-561.(ZHOU Jingrun, FU Jingli. The integrating factor and conservation quantity for constrained Hamilton system[J]. Chinese Quarterly of Mechanics,2018,39(3): 554-561.(in Chinese))
[29]	OLVER P J. Applications of Lie Group to Differential Equations [M]. Berlin: Springer, 2000.
[30]	OLVER P J. Equivalence, Invariants and Symmetry [M]. Cambridge: Cambridge University Press, 1995.
[31]	梅凤翔, 吴润衡, 张永发. 非Чет〖KG-*4〗аев型非完整系统的Lie对称性与守恒量[J]. 力学学报, 1988,30(4): 468-474.(MEI Fengxiang, WU Runheng, ZHANG Yongfa. Lie symmetries and conserved quantities of nonholonomic systems of non-Chetaev’s type[J]. Acta Mechanica Sinica,1988,30(4): 468-474.(in Chinese))
[32]	ZHAO Y Y, MEI F X. On symmetry and invariant of dynamical systems[J]. Advances in Mechanics,1993,23(3): 360-372.
[33]	梅凤翔. 非完整系统力学基础[M]. 北京: 北京工业学院出版社, 1985.(MEI Fengxiang. Nonholonomic Mechanics Foundation [M]. Beijing: Beijing Institute of Technology Press, 1985.(in Chinese))
[34]	XIANG Pei, GUO Yongxin, FU Jingli. Time and space fractional Schrdinger equation with fractional factor[J]. Communications in Theoretical Physics,2019,71(1): 16-26.
[35]	梅凤翔. 约束力学系统的对称性与守恒量[M]. 北京: 北京理工大学出版社, 2004.(MEI Fengxiang. Symmetry and Conserved Quantity of Constrained Mechanical System [M]. Beijing: Beijing Institute of Technology Press, 2004.(in Chinese))
[36]	CHEN X W, LIU C, MEI F X, et al. Conformal invariance and Hojman conserved quantities of first order Lagrange systems[J]. Chinese Physics B,2008,50(9): 3180-3184.
[37]	FU J L, CHEN B Y, FU H, et al. Velocity-dependent symmetries and non-Noether conserved quantities of electromechanical systems[J]. Science China: Physics, Mechanics and Astronomy,2011,54(2): 288-295.
[38]	FU J L, LI X W, LI C R, et al. Symmetries and exact solutions of discrete nonconservative systems[J]. Science China: Physics, Mechanics and Astronomy,2010,53(9): 1699-1706.
[39]	FENG K, QIN M Z. Symplectic Geometric Algorithms for Hamiltonian System [M]. New York: Springer, 2009.
[40]	刘荣万, 傅景礼. 非完整非保守力学系统在相空间的Lie对称性与守恒量[J]. 应用数学和力学, 1999,20(6): 597-601.(LIU Rongwan, FU Jingli. Lie symmetries and conserved quantities of nonconservative nonholonomic system in phase space[J]. Applied Mathematics and Mechanics,1999,20(6): 597-601.(in Chinese))
[41]	乔永芬, 张耀良, 赵淑红. 完整非保守系统Raitzin正则运动方程的积分因子和守恒定理[J]. 物理学报, 2002,51(8): 1661-1665.(QIAO Yongfen, ZHANG Yaoliang, ZHAO Shuhong. Integrating factors and conservation laws for the Raitzins canonical equations of motion of non-conservative dynamical systems[J]. Acta Physica Sinca,2002,51(8): 1661-1665.(in Chinese))
[42]	傅景礼, 刘荣万. 准坐标下非完整力学系统的Lie对称性和守恒量[J]. 数学物理学报, 2000,20(1): 63-69.(FU Jingli, LIU Rongwan. Lie symmetries and conserver quantities of nonholonomic mechanical systems in terms of quasi-coordinates[J]. Acta Mathematica Scientia, 2000,20(1): 63-69.(in Chinese))
[43]	郑明亮. 机械多体系统动力学非线性最优控制问题的Noether理论[J]. 应用数学和力学, 2018,〖STHZ〗 39(7): 776-784.(ZHENG Mingliang. The Noether theorem for nonlinear optimal control problems of mechanical multibody system dynamics[J]. Applied Mathematics and Mechanics,2018,〖STHZ〗 39(7): 776-784.(in Chinese))
[44]	崔新斌, 傅景礼. 汽车电磁悬架系统的Noether对称性及其应用[J]. 应用数学和力学, 2017,38(12): 1331-1341.(CUI Xinbin, FU Jingli. Noether symmetry of automotive electromagnetic suspension systems and its application[J]. Applied Mathematics and Mechanics,2017,38(12): 1331-1341.(in Chinese))
[45]	李子平. 约束哈密顿系统及其对称性质[M]. 北京: 北京工业大学出版社, 1999.(LI Ziping. Constrained Hamiltonian Systems and Their Symmetric Properties [M]. Beijing: Beijing University of Technology Press, 1999.(in Chinese))
[46]	周景润, 傅景礼. 约束Hamilton系统的积分因子和守恒量及其在场论中的应用[J]. 数学物理学报, 2019,39(1): 38-48.(ZHOU Jingrun, FU Jingli. Integrating factors and conserved quantities for constrained Hamilton systems and its applications in field theory[J]. Acta Mathematica Scientia,2019,39(1): 38-48.(in Chinese))
[47]	LI Z P. Symmetry in field theory for singular Lagrangian with derivatives of higher order[J]. Acta Mathematica Scientia,1994,14(S1): 30-39.
[48]	LI Z P. The symmetry transformation of the constrained system with high-order derivatives[J]. Acta Mathematica Scientia,1985,5(4): 379-388.
[49]	ZHANG R L, LIU J D, TANG Y F, et al. Canonicalization and symplectic simulation of gyrocenter dynamics in time-independent magnetic fields[J]. Physics of Plasmas,2014,21(3): 2445-2458
[50]	TANG Y F. Formal energy of a symplectic scheme for Hamiltonian systems and its applications[J]. Computers and Mathematics With Applications,1994,27(7): 31-39.
[51]	ZHNAG R L,TANG Y F, ZHU B B, et al. Convergence analysis of the formal energies of symplectic methods for Hamiltonian systems[J]. Science China Mathematics,2016,59(2): 1-18.

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约束Hamilton系统的Lie对称性及其在场论中的应用

doi: 10.21656/1000-0887.390218

作者简介:
周景润(1988—)，男，硕士(E-mail: 869569521@qq.com);傅景礼(1955—)，男，教授，博士，博士生导师(通讯作者. E-mail: sqfujingli@163.com).

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Lie Symmetry of Constrained Hamiltonian Systems and Its Application in Field Theory

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留言板

约束Hamilton系统的Lie对称性及其在场论中的应用

doi: 10.21656/1000-0887.390218

作者简介: 周景润(1988—)，男，硕士(E-mail: 869569521@qq.com);傅景礼(1955—)，男，教授，博士，博士生导师(通讯作者. E-mail: sqfujingli@163.com).

计量

出版历程

Lie Symmetry of Constrained Hamiltonian Systems and Its Application in Field Theory

计量

出版历程

目录

作者简介:
周景润(1988—)，男，硕士(E-mail: 869569521@qq.com);傅景礼(1955—)，男，教授，博士，博士生导师(通讯作者. E-mail: sqfujingli@163.com).