Caputo导数下分数阶Lagrange系统的Noether准对称性与守恒量
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摘要
利用时间重新参数化方法,研究分数阶Lagrange系统的Noether准对称性与守恒量。首先,导出Caputo导数下的分数阶Lagrange方程。其次,给出分数阶Lagrange系统的分数阶守恒量的定义,在时间不变的特殊无限小变换群下给出分数阶Lagrange系统的Noether准对称性的定义和判据,并建立Noether准对称性定理。然后,利用时间重新参数化方法,给出在时间变化的一般无限小变换群下分数阶Lagrange系统的Noether准对称性的定义和判据,建立Noether准对称性定理。最后,举例说明结果的应用。
The Noether quasi-symmetry and the conserved quantity for a fractional Lagrange system are studied by using the time-reparameterization method. Firstly,the fractional Lagrange equations in terms of Caputo derivatives are derived; Secondly,the definition of fractional conserved quantity for the fractional Lagrange system is given,and based on the special infinitesimal transformations of group without transforming time,the definition and the criterion of the Noether quasi-symmetry for the fractional Lagrange system are given,and Noether' s quasi-symmetry theorem is established; Finally,the definition and the criterion of Noether quasi-symmetry for the fractional Lagrange system under the general infinitesimal transformations of group with transforming time are given,Noether' s quasi-symmetry theorem is derived by using the time-reparameterization method. An example is given to illustrate the application of the results.
The Noether quasi-symmetry and the conserved quantity for a fractional Lagrange system are studied by using the time-reparameterization method. Firstly,the fractional Lagrange equations in terms of Caputo derivatives are derived; Secondly,the definition of fractional conserved quantity for the fractional Lagrange system is given,and based on the special infinitesimal transformations of group without transforming time,the definition and the criterion of the Noether quasi-symmetry for the fractional Lagrange system are given,and Noether' s quasi-symmetry theorem is established; Finally,the definition and the criterion of Noether quasi-symmetry for the fractional Lagrange system under the general infinitesimal transformations of group with transforming time are given,Noether' s quasi-symmetry theorem is derived by using the time-reparameterization method. An example is given to illustrate the application of the results.
引文
[1]RIEWE F.Nonconservative Lagrangian and Hamiltonian mechanics[J].Physical Review E,1996,53(2):1890-1899.
[2]RIEWE F.Mechanics with fractional derivatives[J].Physical Review E,1997,55(3):3581-3592.
[3]FREDERICO G S F,TORRES D F M.A formulation of Noether’s theorem for fractional problems of the calculus of variations[J].Mathematical Analysis and Applications,2007,334(2):834-846.
[4]FREDERICO G S F,TORRES D F M.Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem[J].International Mathematical Forum,2008,3(10):479-493.
[5]FREDERICO G S F,TORRES D F M.Fractional Noether’s theorem in the Riesz-Caputo sense[J].Applied Mathematics and Computation,2010,217(3):1023-1033.
[6]FREDERICO G S F,TORRES D F M.Fractional isoperimetric Noether’s theorem in the Riemann-Liouville sense[J].Reports on Mathematical Physics,2013,71(3):291-304.
[7]ZHANG Y,ZHOU Y.Symmetries and conserved quantities for fractional action-like Pfaff variational problems[J].Nonlinear Dynamics,2013,73(1/2):783-793.
[8]ZHOU Y,ZHANG Y.Noether’s theorems of a fractional Birkhoffian system within Riemann-Liouville derivatives[J].Chinese Physics B,2014,23(12):281-288.
[9]ZHANG Y,ZHAI X H.Noether symmetries and conserved quantities for fractional Birkhoffian systems[J].Nonlinear Dynamics,2015,81(1/2):469-480.
[10]ZHAI X H,ZHANG Y.Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay[J].Communications in Nonlinear Science and Numerical Simulation,2016,36:81-97.
[11]张毅,周燕.基于Riesz导数的分数阶Birkhoff系统的Noether对称性与守恒量[J].北京大学学报(自然科学版),2016,52(4):658-668.ZHOU Y,ZHANG Y.Noether symmetry and conserved quantity for fractional Birkhoffian systems in terms of Riesz derivatives[J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):658-668.
[12]YAN B,ZHANG Y.Noether’s theorem for fractional Birkhoffian systems of variable order[J].Acta Mechanica,2016,227(9):2439-2449.
[13]SONG C J,ZHANG Y.Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems[J].International Journal of Non-Linear Mechanics,2017,90:32-38.
[14]何胜鑫,朱建青,张毅.基于分数阶模型的非保守系统的Noether准对称性[J].中山大学学报(自然科学版),2016,55(2):58-63.HE S X,ZHU J Q,ZHANG Y.Noether quasi-symmetry for non-conservative systems based on fractional model[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2016,55(2):58-63.
[15]何胜鑫,朱建青.基于分数阶模型的相空间中非保守力学系统的Noether准对称性[J].中山大学学报(自然科学版),2015,54(4):37-42.HE S X,ZHU J Q.Noether quasi-symmetry for nonconservative mechanical system in phase space based on fractional models[J].Acta Scientiarum Scientiraum Naturalium Universitatis Sunyatseni,2015,54(4):37-42.
[16]丁金凤,金世欣,张毅.基于Caputo导数下的含时滞的Hamilton系统的分数阶Noether理论[J].中山大学学报(自然科学版),2016,55(6):79-85.DING J F,JING S X,ZHANG Y.Fractional Noether theorems for Hamilton system with time delay based on Caputo derivatives[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2016,55(6):79-85.
[17]刘艳东,张毅.研究Noether准对称性定理的时间重新参数化方法[J].苏州科技大学学报(自然科学版),2017,34(2):1-7.LIU Y D,ZHANG Y.The time-reparameterization method for study of Noether’s quasi-symmetry theorems[J].Journal of Suzhou University of Science and Technology(Natural Science Edition),2017,34(2):1-7.
[18]梅凤翔.经典约束力学系统对称性与守恒量研究进展[J].力学进展,2009,39(1):37-43.MEI F X.Advances in the symmetries and conserved quantities of classical constrained systems[J].Advances in Mechanics,2009,39(1):37-43.
[19]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.
[20]PONDLUBNY I.Fractional differential equations[M].San Diego:Academic Press,1999:62-77.
[21]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and applications of fractional differential equations[M].Amsterdam:Elsevier B V,2006:69-89.
[2]RIEWE F.Mechanics with fractional derivatives[J].Physical Review E,1997,55(3):3581-3592.
[3]FREDERICO G S F,TORRES D F M.A formulation of Noether’s theorem for fractional problems of the calculus of variations[J].Mathematical Analysis and Applications,2007,334(2):834-846.
[4]FREDERICO G S F,TORRES D F M.Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem[J].International Mathematical Forum,2008,3(10):479-493.
[5]FREDERICO G S F,TORRES D F M.Fractional Noether’s theorem in the Riesz-Caputo sense[J].Applied Mathematics and Computation,2010,217(3):1023-1033.
[6]FREDERICO G S F,TORRES D F M.Fractional isoperimetric Noether’s theorem in the Riemann-Liouville sense[J].Reports on Mathematical Physics,2013,71(3):291-304.
[7]ZHANG Y,ZHOU Y.Symmetries and conserved quantities for fractional action-like Pfaff variational problems[J].Nonlinear Dynamics,2013,73(1/2):783-793.
[8]ZHOU Y,ZHANG Y.Noether’s theorems of a fractional Birkhoffian system within Riemann-Liouville derivatives[J].Chinese Physics B,2014,23(12):281-288.
[9]ZHANG Y,ZHAI X H.Noether symmetries and conserved quantities for fractional Birkhoffian systems[J].Nonlinear Dynamics,2015,81(1/2):469-480.
[10]ZHAI X H,ZHANG Y.Noether symmetries and conserved quantities for fractional Birkhoffian systems with time delay[J].Communications in Nonlinear Science and Numerical Simulation,2016,36:81-97.
[11]张毅,周燕.基于Riesz导数的分数阶Birkhoff系统的Noether对称性与守恒量[J].北京大学学报(自然科学版),2016,52(4):658-668.ZHOU Y,ZHANG Y.Noether symmetry and conserved quantity for fractional Birkhoffian systems in terms of Riesz derivatives[J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):658-668.
[12]YAN B,ZHANG Y.Noether’s theorem for fractional Birkhoffian systems of variable order[J].Acta Mechanica,2016,227(9):2439-2449.
[13]SONG C J,ZHANG Y.Conserved quantities and adiabatic invariants for fractional generalized Birkhoffian systems[J].International Journal of Non-Linear Mechanics,2017,90:32-38.
[14]何胜鑫,朱建青,张毅.基于分数阶模型的非保守系统的Noether准对称性[J].中山大学学报(自然科学版),2016,55(2):58-63.HE S X,ZHU J Q,ZHANG Y.Noether quasi-symmetry for non-conservative systems based on fractional model[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2016,55(2):58-63.
[15]何胜鑫,朱建青.基于分数阶模型的相空间中非保守力学系统的Noether准对称性[J].中山大学学报(自然科学版),2015,54(4):37-42.HE S X,ZHU J Q.Noether quasi-symmetry for nonconservative mechanical system in phase space based on fractional models[J].Acta Scientiarum Scientiraum Naturalium Universitatis Sunyatseni,2015,54(4):37-42.
[16]丁金凤,金世欣,张毅.基于Caputo导数下的含时滞的Hamilton系统的分数阶Noether理论[J].中山大学学报(自然科学版),2016,55(6):79-85.DING J F,JING S X,ZHANG Y.Fractional Noether theorems for Hamilton system with time delay based on Caputo derivatives[J].Acta Scientiarum Naturalium Universitatis Sunyatseni,2016,55(6):79-85.
[17]刘艳东,张毅.研究Noether准对称性定理的时间重新参数化方法[J].苏州科技大学学报(自然科学版),2017,34(2):1-7.LIU Y D,ZHANG Y.The time-reparameterization method for study of Noether’s quasi-symmetry theorems[J].Journal of Suzhou University of Science and Technology(Natural Science Edition),2017,34(2):1-7.
[18]梅凤翔.经典约束力学系统对称性与守恒量研究进展[J].力学进展,2009,39(1):37-43.MEI F X.Advances in the symmetries and conserved quantities of classical constrained systems[J].Advances in Mechanics,2009,39(1):37-43.
[19]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.
[20]PONDLUBNY I.Fractional differential equations[M].San Diego:Academic Press,1999:62-77.
[21]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and applications of fractional differential equations[M].Amsterdam:Elsevier B V,2006:69-89.
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