基于El-Nabulsi模型的一类非完整系统的积分因子与守恒量
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摘要
利用积分因子方法研究一类非完整系统的守恒量.基于按周期律拓展的分数阶积分的El-Nabulsi模型,给出了一类非完整系统部分正则形式的运动微分方程;定义了该系统的运动微分方程的积分因子;利用积分因子方法构建该系统的守恒量,建立了系统的守恒定理和逆定理,并给出求解积分因子的广义Killing方程.最后举例说明结果的应用.
The conserved quantity of a class of nonholonomic systems was studied by integrating factor method.Under the El-Nabulsi model that was based on a fractional integral extended by periodic laws,the differential equations of motion with partial canonical form for a class of nonholonomic system was given.The integrating factor for the canonical equation of the system was defined.The conserved quantity of the system was constructed by integrating factor method,the conservation theorem and inverse theorem of the system were established,and the generalized Killing equation of integrating factor was given.Finally,an example was given to illustrate the application of the results.
The conserved quantity of a class of nonholonomic systems was studied by integrating factor method.Under the El-Nabulsi model that was based on a fractional integral extended by periodic laws,the differential equations of motion with partial canonical form for a class of nonholonomic system was given.The integrating factor for the canonical equation of the system was defined.The conserved quantity of the system was constructed by integrating factor method,the conservation theorem and inverse theorem of the system were established,and the generalized Killing equation of integrating factor was given.Finally,an example was given to illustrate the application of the results.
引文
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[9] ZHANG Y.A general approach to the construction of conservation laws for Birkhoffian systems in event space[J].Communications in Theoretical Physics,2008,50(10):851-854.
[10]束方平,张毅.用积分因子方法研究广义Birkhoff系统的守恒律[J].华中师范大学学报(自然科学版),2014,48(1):42-45.SHU F P,ZHANG Y.Integrating factors and conservation laws for a generalized Birkhoffian system[J].Journal of Central China Normal University(Natural Sciences),2014,48(1):42-45.(Ch).
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[12]束方平,张毅,朱建青.基于分数阶模型的Lagrange系统的积分因子与守恒量[J].苏州科技学院学报(自然科学版),2015,32(2):1-5.SHU F P,ZHANG Y,ZHU J Q.Integrating factors and conserved quantities for Lagrange systems based on fractional order model[J].Journal of Suzhou University of Science and Technology(Natural Science),2015,32(2):1-5.(Ch).
[13] RIEWE F.Nonconservative Lagrangian and Hamiltonian mechanics[J]. Physical Review E,1996,53(2):1890-1899.
[14] RIEWE F.Mechanics with fractional derivatives[J].Physical Review E,1997,55(3):3581-3592.
[15] EL-NABULSI R A.A fractional approach of nonconservative Lagrangian dynamics[J].Fizika A,2005,14(4):289-298.
[16] EL-NABULSI R A.Fractional variational problems from extended exponentially fractional integral[J]. Applied Mathematics&Computation,2011,217(22):9492-9496.
[17] EL-NABULSI R A.A periodic functional approach to the calculus of variations and the problem of time-dependent damped harmonic oscillators[J].Applied Mathematics Letters,2011,24(10):1647-1653.
[18]丁金凤,张毅.基于按周期律拓展的分数阶积分的ElNabulsi-Pfaff变分问题的Noether对称性[J].华中师范大学学报(自然科学版),2014,48(6):829-833.DING J F,ZHANG Y.Noether symmetries for El-NabulsiPfaff variational problem based on fractional integral extended by periodic law[J].Journal of Central China Normal University(Natural Sciences),2014,48(6):829-833.(Ch).
[19] LONG Z X,ZHANG Y.Noether's theorem for non-conservative Hamilton system based on EI-Nabulsi dynamical model extended by periodic laws[J].Chinese Physics B,2014,23(11):114501.
[20] LONG Z X,ZHANG Y.Noether's theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space[J].Acta Mechanica,2014,225(1):77-90.
[21] SONG C J,ZHANG Y.Conserved quantities and adiabatic invariants for El-Nabulsi's fractional Birkhoff system[J].International Journal of Theoretical Physics,2015,54(8):2481-2493.
[22]陈菊,张毅.El-Nabulsi动力学模型下非Chetaev型非完整系统的精确不变量与绝热不变量[J].物理学报,2015,64(3):034502.CHEN J,ZHANG Y.Exact invariants and adiabatic invariants for nonholonomic systems in non-Chetaev's type based on El-Nabulsi dynamical models[J].Acta Physica Sinica,2015,64(3):034502.(Ch).
[23]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.(Ch).
[2]朱照宣,周启钊,殷金生.理论力学[M].北京:北京大学出版社,1982.ZHU Z X,ZHOU Q Z,YIN J S.Theoretical Mechanics[M].Beijing:Peking University Press,1982.(Ch).
[3] DJUKI C'D S,SUTELA T.Integrating factors and conservation laws for nonconservative dynamical systems[J].International Journal of Non-Linear Mechanics,1984,19(4):331-339.
[4] QIAO Y F,ZHANG Y L,HAN G C.Integrating factors and conservation theorem for holonomic nonconservative dynamical systems in generalized classical mechanics[J].Chinese Physics,2002,11(10):988-992.
[5]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 Y F,ZHANG Y L,ZHAO S H.Integrating factors and conservation laws for the Raitzin's canonical equations of motion of nonconservative dynamical systems[J].Acta Physica Sinica,2002,51(8):1661-1665.(Ch).
[6]乔永芬,张耀良,韩广才.非完整保守系统Raitzin正则运动方程的积分因子和守恒定理[J].兵工学报,2003,24(2):162-166.QIAO Y F,ZHANG Y L,HAN G C.Integrating factors and conservation theorem for Raitzin's canonical equations of motion of nonholonomic nonconservative dynamical systems[J].Acta Armamentarii,2003,24(2):162-166.(Ch).
[7]张毅,葛伟宽.用积分因子方法研究非完整约束系统的守恒律[J].物理学报,2003,52(10):2363-2367.ZHANG Y,GE W K.Integrating factors and conservation laws for non-holonomic dynamical systems[J].Acta Physica Sinca,2003,52(10):2363-2367.(Ch).
[8]张毅,薛纭.Birkhoff系统的积分因子与守恒定理[J].力学季刊,2003,24(2):280-285.ZHANG Y,XUE Y.Integrating factors and conservation theorems for Birkhoffian systems[J].Chinese Quarterly of Mechanics,2003,24(2):280-285.(Ch).
[9] ZHANG Y.A general approach to the construction of conservation laws for Birkhoffian systems in event space[J].Communications in Theoretical Physics,2008,50(10):851-854.
[10]束方平,张毅.用积分因子方法研究广义Birkhoff系统的守恒律[J].华中师范大学学报(自然科学版),2014,48(1):42-45.SHU F P,ZHANG Y.Integrating factors and conservation laws for a generalized Birkhoffian system[J].Journal of Central China Normal University(Natural Sciences),2014,48(1):42-45.(Ch).
[11] ZHANG Y.Conservation laws for mechanical systems with unilateral holonomic constraints[J].Progress in Natural Science,2004,14(1):55-59.
[12]束方平,张毅,朱建青.基于分数阶模型的Lagrange系统的积分因子与守恒量[J].苏州科技学院学报(自然科学版),2015,32(2):1-5.SHU F P,ZHANG Y,ZHU J Q.Integrating factors and conserved quantities for Lagrange systems based on fractional order model[J].Journal of Suzhou University of Science and Technology(Natural Science),2015,32(2):1-5.(Ch).
[13] RIEWE F.Nonconservative Lagrangian and Hamiltonian mechanics[J]. Physical Review E,1996,53(2):1890-1899.
[14] RIEWE F.Mechanics with fractional derivatives[J].Physical Review E,1997,55(3):3581-3592.
[15] EL-NABULSI R A.A fractional approach of nonconservative Lagrangian dynamics[J].Fizika A,2005,14(4):289-298.
[16] EL-NABULSI R A.Fractional variational problems from extended exponentially fractional integral[J]. Applied Mathematics&Computation,2011,217(22):9492-9496.
[17] EL-NABULSI R A.A periodic functional approach to the calculus of variations and the problem of time-dependent damped harmonic oscillators[J].Applied Mathematics Letters,2011,24(10):1647-1653.
[18]丁金凤,张毅.基于按周期律拓展的分数阶积分的ElNabulsi-Pfaff变分问题的Noether对称性[J].华中师范大学学报(自然科学版),2014,48(6):829-833.DING J F,ZHANG Y.Noether symmetries for El-NabulsiPfaff variational problem based on fractional integral extended by periodic law[J].Journal of Central China Normal University(Natural Sciences),2014,48(6):829-833.(Ch).
[19] LONG Z X,ZHANG Y.Noether's theorem for non-conservative Hamilton system based on EI-Nabulsi dynamical model extended by periodic laws[J].Chinese Physics B,2014,23(11):114501.
[20] LONG Z X,ZHANG Y.Noether's theorem for fractional variational problem from El-Nabulsi extended exponentially fractional integral in phase space[J].Acta Mechanica,2014,225(1):77-90.
[21] SONG C J,ZHANG Y.Conserved quantities and adiabatic invariants for El-Nabulsi's fractional Birkhoff system[J].International Journal of Theoretical Physics,2015,54(8):2481-2493.
[22]陈菊,张毅.El-Nabulsi动力学模型下非Chetaev型非完整系统的精确不变量与绝热不变量[J].物理学报,2015,64(3):034502.CHEN J,ZHANG Y.Exact invariants and adiabatic invariants for nonholonomic systems in non-Chetaev's type based on El-Nabulsi dynamical models[J].Acta Physica Sinica,2015,64(3):034502.(Ch).
[23]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.(Ch).