基于非标准Lagrange函数的动力学系统的Lie对称性与Mei对称性
|
摘要
提出并研究在非标准Lagrange函数下动力学系统的Lie对称性与Mei对称性.基于系统的Lagrange方程,引入无限小变换及其生成元向量,给出了Lie对称性和Mei对称性的定义,建立了两类非标准Lagrange函数(指数Lagrange函数和幂律Lagrange函数)下动力学系统的Lie对称性结构方程和Mei对称性结构方程,导出了Lie对称性导致的Noether守恒量和Mei对称性导致的Mei守恒量,并结合算例说明结果的应用.
The Lie symmetry and the Mei symmetry for dynamical systems with non-standard Lagrangians have been presented and studied.By introducing the infinitesimal transformations and their generating vectors,and in light of the Lagrange equations of the systems,the definitions of Lie symmetry and Mei symmetry have been given,and the structure equations of Lie symmetry and Mei symmetry for two types of dynamical systems with non-standard Lagrangians(i.e.with exponential Lagrangians and power-law Lagrangians) have been established.The Noether conserved quantity resulted from Lie symmetry and the Mei conserved quantity resulted from Mei symmetry have been derived.In addition four examples have been given to illustrate the application of the results.
The Lie symmetry and the Mei symmetry for dynamical systems with non-standard Lagrangians have been presented and studied.By introducing the infinitesimal transformations and their generating vectors,and in light of the Lagrange equations of the systems,the definitions of Lie symmetry and Mei symmetry have been given,and the structure equations of Lie symmetry and Mei symmetry for two types of dynamical systems with non-standard Lagrangians(i.e.with exponential Lagrangians and power-law Lagrangians) have been established.The Noether conserved quantity resulted from Lie symmetry and the Mei conserved quantity resulted from Mei symmetry have been derived.In addition four examples have been given to illustrate the application of the results.
引文
[1]NOETHER A E.Invariante variations probleme[J].Nachr Akad Wiss G9ttingen.Math Phys,1918,KI,II:235-257.
[2]LUTZKY M.Dynamical symmetries and conserved quantities[J].Journal of Physics A:Mathematical and General,1979,12(7):973-981.
[3]MEI F X.Form invariance of Lagrange system[J].Journal of Beijing Institute of Technology,2000,9(2):120-124.
[4]赵跃宇.非保守力学系统的Lie对称性与守恒量[J].力学学报,1994,30(4):380-384.ZHAO Y Y.Conservative quantities and Lie’s symmetries of nonconservative dynamical systems[J].Acta Mechanica Sinica,1994,30(4):380-384.
[5]贾利群,郑世旺,张耀宇.事件空间中非Chetaev型非完整系统的Mei对称性与Mei守恒量[J].物理学报,2007,56(10):5 575-5 579.JIA L Q,ZHENG S W,ZHANG Y Y.Mei symmetry and Mei conserved quantity of nonholonomic systems of non-Chetaev’s type in event space[J].Acta Physica Sinica,2007,56(10):5 575-5 579.
[6]贾利群,罗绍凯,张耀宇.非完整系统Nielsen方程Mei对称性与Mei守恒量[J].物理学报,2008,57(4):2 006-2 010.JIA L Q,LUO S K,ZHANG Y Y.Mei symmetry and Mei conserved quantity of Nielsen equation for a nonholonomic system[J].Acta Physica Sinica,2008,57(4):2 006-2 010.
[7]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 F X.Applications of Lie Groups and Lie algebras to constrained mechanical systems[M].Beijing:Science Press,1999.
[8]罗绍凯.Hamilton系统的Mei对称性、Noether对称性和Lie对称性[J].物理学报,2003,52(12):2 941-2 944.LUO S K.Mei symmetry,Noether symmetry and Lie symmetry of Hamiltonian system[J].Acta Physica Sinica,2003,52(12):2 941-2 944.
[9]ZHANG Y,XUE Y.Conserved quantities from Lie symmetries for nonholonomic systems[J].Journal of Southeast University,2003,19(3):289-292.
[10]ZHANG Y.Generalized Lutzky conserved quantities of holonomic systems with remainder coordinates subjected to unilateral constraints[J].Communication in Theoretical Physics,2006,45(4):732-736.
[11]张毅.广义经典力学系统的对称性与Mei守恒量[J].物理学报,2005,54(7):2 980-2 984.ZHANG Y.Symmetries and Mei conserved quantities for systems of generalized classical mechanics[J].Acta Physica Sinica,2005,54(7):2 980-2 984.
[12]张毅,葛伟宽.相对论性力学系统的Mei对称性导致的新守恒律[J].物理学报,2005,54(4):1 464-1 467.ZHANG Y,GE W K.A new conservation law from Mei symmetry for the relativistic mechanical system[J].Acta Physica Sinica,2005,54(4):1 464-1 467.
[13]方建会,彭勇,廖永潘.关于Lagrange系统和Hamilton系统的Mei对称性[J].物理学报,2005,54(2):496-499.FANG J H,PENG Y,LIAO Y P.On Mei symmetry of Lagrangian system and Hamiltonian system[J].Acta Physica Sinica,2005,54(2):496-499.
[14]方建会,廖永潘,彭勇.相空间中力学系统的两类Mei对称性及守恒量[J].物理学报,2005,54(2):500-503.FANG J H,LIAO Y P,PENG Y.Tow kinds of Mei symmeties and conserved quantities of a mechanical system in phase space[J].Acta Physica Sinica,2005,54(2):500-503.
[15]张晔,张毅,陈向炜.事件空间中Birkhoff系统的Mei对称性与守恒量[J].云南大学学报:自然科学版,2016,38(3):406-411.ZHANG Y,ZHANG Y,CHEN X W.Mei symmetry and conserved quantity of a Birkhoffian system in event space[J].Journal of Yunnan University:Natural Sciences Edition,2016,38(3):406-411.
[16]梅凤翔,尚玫.一阶Lagrange系统的Lie对称性与守恒量[J].物理学报,2000,49(10):1 901-1 903.MEI F X,SHANG M.Lie symmetries and conserved quantities of first order Lagrange systems[J].Acta Physica Sinica,2000,49(10):1 901-1 903.
[17]HOJMAN S A.A new conservation law constructed without using either Lagrangians or Hamiltonians[J].Journal of Physics A:Mathematical and General,1992,25(7):L291-L295.
[18]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 F X.Symmetries and Conserved Quantities of Constrained Mechanical Systems[M].Beijing:Beijing Institute of Technology Press,2004.
[19]MUSIELAK Z E.Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients[J].Journal of Physics A:Mathematical and Theoretical,2008,41(5):055205.
[20]MUSIELAK Z E.General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems[J].Chaos,Solitons&Fractals,2009,42(5):2 645-2 652.
[21]EL-NABULSI A R.Non-linear dynamics with non-standard Lagrangians[J].Qualitative Theory of Dynamical Systems,2013,12(2):273-291.
[22]EL-NABULSI A R.Non-standard fractional Lagrangians[J].Nonlinear Dynamics,2013,74(1/2):381-394.
[23]EL-NABULSI A R.Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent[J].Computational and Applied Mathematics,2014,33(1):163-179.
[24]DIMITRIJEVIC D D,MILOSEVIC M.About nonstandard Lagrangians in cosmology[C]//Poceedings of the Physics Conference TIM-11.AIP Publishing,2012,1472(1):41-46.
[25]CARINENA J F,RANADA M F,SANTANDER M.Lagrangian formalism for nonlinear second-order Riccati systems:one-dimensional integrability and two-dimensional superintegrability[J].Journal of Mathematical Physics,2005,46(6):062703.
[26]CHANDRASEKAR V K,PANDEY S N,SENTHILVELAN M,et al.A simple and unified approach to identify integrable nonlinear oscillators and systems[J].Journal of Mathematical Physics,2006,47(2):023508.
[27]ALEKSEEV A I,ARBUZOV B A.Classical Yang-Mills field theory with nonstandard Lagrangians[J].Theoretical and Mathematical Physics,1984,59(1):372-378.
[28]ZHANG Y,ZHOU X S.Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians[J].Nonlinear Dynamics,2016,84(4):1 867-1 876.
[29]周小三,张毅.基于非标准Lagrange函数的动力学系统的Routh降阶法[J].力学季刊,2016,37(1):15-21.ZHOU X S,ZHANG Y.Routh method of reduction for dynamic systems with non-standard Lagrangians[J].Chinese Quarterly of Mechanics,2016,37(1):15-21.
[2]LUTZKY M.Dynamical symmetries and conserved quantities[J].Journal of Physics A:Mathematical and General,1979,12(7):973-981.
[3]MEI F X.Form invariance of Lagrange system[J].Journal of Beijing Institute of Technology,2000,9(2):120-124.
[4]赵跃宇.非保守力学系统的Lie对称性与守恒量[J].力学学报,1994,30(4):380-384.ZHAO Y Y.Conservative quantities and Lie’s symmetries of nonconservative dynamical systems[J].Acta Mechanica Sinica,1994,30(4):380-384.
[5]贾利群,郑世旺,张耀宇.事件空间中非Chetaev型非完整系统的Mei对称性与Mei守恒量[J].物理学报,2007,56(10):5 575-5 579.JIA L Q,ZHENG S W,ZHANG Y Y.Mei symmetry and Mei conserved quantity of nonholonomic systems of non-Chetaev’s type in event space[J].Acta Physica Sinica,2007,56(10):5 575-5 579.
[6]贾利群,罗绍凯,张耀宇.非完整系统Nielsen方程Mei对称性与Mei守恒量[J].物理学报,2008,57(4):2 006-2 010.JIA L Q,LUO S K,ZHANG Y Y.Mei symmetry and Mei conserved quantity of Nielsen equation for a nonholonomic system[J].Acta Physica Sinica,2008,57(4):2 006-2 010.
[7]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 F X.Applications of Lie Groups and Lie algebras to constrained mechanical systems[M].Beijing:Science Press,1999.
[8]罗绍凯.Hamilton系统的Mei对称性、Noether对称性和Lie对称性[J].物理学报,2003,52(12):2 941-2 944.LUO S K.Mei symmetry,Noether symmetry and Lie symmetry of Hamiltonian system[J].Acta Physica Sinica,2003,52(12):2 941-2 944.
[9]ZHANG Y,XUE Y.Conserved quantities from Lie symmetries for nonholonomic systems[J].Journal of Southeast University,2003,19(3):289-292.
[10]ZHANG Y.Generalized Lutzky conserved quantities of holonomic systems with remainder coordinates subjected to unilateral constraints[J].Communication in Theoretical Physics,2006,45(4):732-736.
[11]张毅.广义经典力学系统的对称性与Mei守恒量[J].物理学报,2005,54(7):2 980-2 984.ZHANG Y.Symmetries and Mei conserved quantities for systems of generalized classical mechanics[J].Acta Physica Sinica,2005,54(7):2 980-2 984.
[12]张毅,葛伟宽.相对论性力学系统的Mei对称性导致的新守恒律[J].物理学报,2005,54(4):1 464-1 467.ZHANG Y,GE W K.A new conservation law from Mei symmetry for the relativistic mechanical system[J].Acta Physica Sinica,2005,54(4):1 464-1 467.
[13]方建会,彭勇,廖永潘.关于Lagrange系统和Hamilton系统的Mei对称性[J].物理学报,2005,54(2):496-499.FANG J H,PENG Y,LIAO Y P.On Mei symmetry of Lagrangian system and Hamiltonian system[J].Acta Physica Sinica,2005,54(2):496-499.
[14]方建会,廖永潘,彭勇.相空间中力学系统的两类Mei对称性及守恒量[J].物理学报,2005,54(2):500-503.FANG J H,LIAO Y P,PENG Y.Tow kinds of Mei symmeties and conserved quantities of a mechanical system in phase space[J].Acta Physica Sinica,2005,54(2):500-503.
[15]张晔,张毅,陈向炜.事件空间中Birkhoff系统的Mei对称性与守恒量[J].云南大学学报:自然科学版,2016,38(3):406-411.ZHANG Y,ZHANG Y,CHEN X W.Mei symmetry and conserved quantity of a Birkhoffian system in event space[J].Journal of Yunnan University:Natural Sciences Edition,2016,38(3):406-411.
[16]梅凤翔,尚玫.一阶Lagrange系统的Lie对称性与守恒量[J].物理学报,2000,49(10):1 901-1 903.MEI F X,SHANG M.Lie symmetries and conserved quantities of first order Lagrange systems[J].Acta Physica Sinica,2000,49(10):1 901-1 903.
[17]HOJMAN S A.A new conservation law constructed without using either Lagrangians or Hamiltonians[J].Journal of Physics A:Mathematical and General,1992,25(7):L291-L295.
[18]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 F X.Symmetries and Conserved Quantities of Constrained Mechanical Systems[M].Beijing:Beijing Institute of Technology Press,2004.
[19]MUSIELAK Z E.Standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients[J].Journal of Physics A:Mathematical and Theoretical,2008,41(5):055205.
[20]MUSIELAK Z E.General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems[J].Chaos,Solitons&Fractals,2009,42(5):2 645-2 652.
[21]EL-NABULSI A R.Non-linear dynamics with non-standard Lagrangians[J].Qualitative Theory of Dynamical Systems,2013,12(2):273-291.
[22]EL-NABULSI A R.Non-standard fractional Lagrangians[J].Nonlinear Dynamics,2013,74(1/2):381-394.
[23]EL-NABULSI A R.Fractional oscillators from non-standard Lagrangians and time-dependent fractional exponent[J].Computational and Applied Mathematics,2014,33(1):163-179.
[24]DIMITRIJEVIC D D,MILOSEVIC M.About nonstandard Lagrangians in cosmology[C]//Poceedings of the Physics Conference TIM-11.AIP Publishing,2012,1472(1):41-46.
[25]CARINENA J F,RANADA M F,SANTANDER M.Lagrangian formalism for nonlinear second-order Riccati systems:one-dimensional integrability and two-dimensional superintegrability[J].Journal of Mathematical Physics,2005,46(6):062703.
[26]CHANDRASEKAR V K,PANDEY S N,SENTHILVELAN M,et al.A simple and unified approach to identify integrable nonlinear oscillators and systems[J].Journal of Mathematical Physics,2006,47(2):023508.
[27]ALEKSEEV A I,ARBUZOV B A.Classical Yang-Mills field theory with nonstandard Lagrangians[J].Theoretical and Mathematical Physics,1984,59(1):372-378.
[28]ZHANG Y,ZHOU X S.Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians[J].Nonlinear Dynamics,2016,84(4):1 867-1 876.
[29]周小三,张毅.基于非标准Lagrange函数的动力学系统的Routh降阶法[J].力学季刊,2016,37(1):15-21.ZHOU X S,ZHANG Y.Routh method of reduction for dynamic systems with non-standard Lagrangians[J].Chinese Quarterly of Mechanics,2016,37(1):15-21.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |