分数阶Lagrange系统的共形不变性与守恒量
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摘要
研究Riemann-Liouville导数下分数阶Lagrange系统的共形不变性与守恒量.首先,建立分数阶d′Alembert-Lagrange原理和分数阶Lagrange方程,给出分数阶Lagrange系统的共形不变性的定义及其确定方程;其次,通过研究分数阶Lagrange系统共形不变性和Lie对称性之间的关系,导出共形因子的表达式;最后,给出相应于分数阶Lagrange系统的共形不变性的Noether型分数阶守恒量.文末,给出算例以说明结果的应用.
The conformal invariance and conserved quantity for afractional Lagrange system in terms of Riemann-Liouville derivatives have been studied. Firstly, the fractional d′Alembert-Lagrange principle and the fractional Lagrange equations have been established, and the definition and corresponding determining equation of conformal invariance for the fractional Lagrange system have been given. Secondly, by studying the relationship between the conformal invariance and the Lie symmetry of the fractional Lagrange system, the conformal factor had derived. Finally, the fractional conserved quantity of Noether type corresponding to the conformal invariance of the fractional Lagrange system has been given. At the end of the paper, an example has been given to illustrate the application.
The conformal invariance and conserved quantity for afractional Lagrange system in terms of Riemann-Liouville derivatives have been studied. Firstly, the fractional d′Alembert-Lagrange principle and the fractional Lagrange equations have been established, and the definition and corresponding determining equation of conformal invariance for the fractional Lagrange system have been given. Secondly, by studying the relationship between the conformal invariance and the Lie symmetry of the fractional Lagrange system, the conformal factor had derived. Finally, the fractional conserved quantity of Noether type corresponding to the conformal invariance of the fractional Lagrange system has been given. At the end of the paper, an example has been given to illustrate the application.
引文
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[24]蔡建乐.一般完整系统Mei对称性的共形不变性与守恒量[J].物理学报,2009,58(1):22-27.DOI:10.3321/j.issn:1000-6737.2009.01.004.Cai J L.Conformal invariance and conserved quantities of Mei symmetry for Lagrange systems[J].Acta Physica Sinica,2009,58(1):22-27.
[25]Zhang Y.Conformal invariance and Noether symmetry,Lie symmetry of Birkhoffian systems in the event space[J].Communications in Theoretical Physics,2010,53(1):166-170.DOI:10.1088/0253-6102/53/1/34.
[26]Chen R,Xu X J.Conformal invariance,Noether symmetry,Lie symmetry and conserved quantities of Hamilton systems[J].Chinese Physics B,2012,21(9):094 501.DOI:10.1088/1674-1056/21/9/094501.
[27]Mei F X,Xie J F,Gang T Q.A conformal invariance for generalized Birkhoff equations[J].Acta Mechanica Sinica,2008,24(5):583-585.DOI:10.1007/s10409-008-0176-8.
[28]Cai J L.Conformal invariance of Mei symmetry for the non-holonomic systems of non-Chetaev’s type[J].Nonlinear Dynamics,2011,69(1/2):487-493.
[29]Luo S K,Dai Y,Zhang X T,et al.Fractional conformal invariance method for finding conserved quantities of dynamical systems[J].International Journal of Non-Linear Mechanics,2017,97(12):107-114.
[30]Luo S K,Dai Y,Zhang X T,et al.Basic theory of fractional conformal invariance of Mei symmetry and its applications to physics[J].International Journal of Theoretical Physics,2018,57(4):1 024-1 038.DOI:10.1007/s10773-017-3635-9.
[31]李彦敏.变质量非完整力学系统的共形不变性[J].云南大学学报:自然科学版,2010,32(1):52-57.Li Y M.Conformal invariance for nonholonomic mechanical systems with variable mass[J].Journal of Yunnan University:Natural Sciences Edition,2010,32(1):52-57.
[32]郑世旺.单面完整约束系统Tzénoff方程Mei对称性的共形不变性与守恒量[J].云南大学学报:自然科学版,2018,40(1):74-81.Zheng S W.On conformal invariance and conserved quantity of Mei symmetry for Tzénoff equation in unilateral constraint system[J].Journal of Yunnan University:Natural Sciences Edition,2018,40(1):74-81.
[33]张毅,梅凤翔.基于Riesz分数阶导数的分数阶运动微分方程[J].北京理工大学学报,2012,32(7):766-770.DOI:10.3969/j.issn.1001-0645.2012.07.023.Zhang Y,Mei F X.Fractional differential equations of motion in terms of Riesz fractional derivatives[J].Transactions of Beijing Institute of Technology,2012,32(7):766-770.
[2]Miller K S,Ross B.An introduction to the fractional calculus and fractional differential equations[M].New York:Wiley Inc.,1993.
[3]Podlubny I.Fractional differential equations[M].San Diego:Academic Press,1999.
[4]Kilbas A A,Srivastava H M,Trujillo J J.Theory and applications of fractional differential equations[M].Amsterdam:Elsevier B V,2006.
[5]Riewe F.Nonconservative Lagrangian and Hamiltonian mechanics[J].Physical Review E,1996,53(2):1 890-1 899.DOI:10.1103/PhysRevE.53.1890.
[6]Riewe F.Mechanics with fractional derivatives[J].Physical Review E,1997,55(3):3 581-3 592.DOI:10.1103/PhysRevE.55.3581.
[7]Noether A E.Invariante variationsprobleme[J].Nachr Akad Wiss Gottingen Math Phys,1918,1(2):235-257.
[8]Frederico G S F,Torres D F M.A formulation of Noether's theorem for fractional problems of the calculus of variations[J].Journal of Mathematical Analysis and Applications,2007,334(2):834-846.DOI:10.1016/j.jmaa.2007.01.013.
[9]Frederico G S F.Fractional optimal control in the sense of Caputo and the fractional Noether's theorem[J].International Mathematical Forum,2008,3(10):479-493.
[10]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):1 023-1 033.DOI:10.1016/j.amc.2010.01.100.
[11]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.DOI:10.1016/S0034-4877(13)60034-8.
[12]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.
[13]Zhang Y,Zhai X H.Noether symmetries and conserved quantities for fractional Birkhoffian systems[J].Nonlinear Dynamics,2015,81(1/2):469-480.DOI:10.1007/s11071-015-2005-5.
[14]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.DOI:10.1016/j.cnsns.2015.11.020.
[15]Zhang S H,Chen B Y,Fu J L.Hamilton formalism and Noether symmetry for mechanico-electrical systems with fractional derivatives[J].Chinese Physics B,2012,21(10):100 202.DOI:10.1088/1674-1056/21/10/100202.
[16]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.DOI:10.1016/j.ijnonlinmec.2017.01.003.
[17]张毅.Caputo导数下分数阶Birkhoff系统的准对称性与分数阶Noether定理[J].力学学报,2017,49(3):693-702.Zhang Y.Quasi-symmetry and Noether’s theorem for fractional Birkhoffian systems in terms of Caputo derivatives[J].Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):693-702.
[18]Zhou S,Fu H,FU J L.Symmetry theories of Hamiltonian systems with fractional derivatives[J].Science China:Physics,Mechanics and Astronomy,2011,54(10):1 847-1 853.DOI:10.1007/s11433-011-4467-x.
[19]Sun Y,Chen B Y,Fu J L.Lie symmetry theorem of fractional nonholonomic systems[J].Chinese Physics B,2014,23(11):110 201.DOI:10.1088/1674-1056/23/11/110201.
[20]Fu J L,Fu L P,Chen B Y,et al.Lie symmetries and their inverse problems of nonholonomic Hamilton systems with fractional derivatives[J].Physics Letters A,2016,380(1/2):15-21.DOI:10.1016/j.physleta.2015.10.002.
[21]张毅.分数阶Birkhoff系统的Lie对称性与守恒量[J].苏州科技大学学报:自然科学版,2017,34(1):1-7.DOI:10.3969/j.issn.1672-0687.2017.01.001.Zhang Y.Lie symmetry and conserved quantity for fractional Birkhoffian system[J].Journal of Suzhou University of Science and Technology:Natural Science Edition,2017,34(1):1-7.
[22]Galiullin A S,Gafarov G G,Malaishka R P,et al.Analytical dynamics of Helmholtz Birkhoff and Nambu systems[M].Moscow:UFN,1997.(in Russian)
[23]刘畅,梅凤翔,郭永新.Lagrange系统的共形不变性与Hojman守恒量[J].物理学报,2008,57(11):6 704-6 708.DOI:10.7498/aps.57.6704.Liu C,Mei F X,Guo Y X.Conformal symmetry and Hojman conserved quantity of Lagrange system[J].Acta Physica Sinica,2008,57(11):6 704-6 708.
[24]蔡建乐.一般完整系统Mei对称性的共形不变性与守恒量[J].物理学报,2009,58(1):22-27.DOI:10.3321/j.issn:1000-6737.2009.01.004.Cai J L.Conformal invariance and conserved quantities of Mei symmetry for Lagrange systems[J].Acta Physica Sinica,2009,58(1):22-27.
[25]Zhang Y.Conformal invariance and Noether symmetry,Lie symmetry of Birkhoffian systems in the event space[J].Communications in Theoretical Physics,2010,53(1):166-170.DOI:10.1088/0253-6102/53/1/34.
[26]Chen R,Xu X J.Conformal invariance,Noether symmetry,Lie symmetry and conserved quantities of Hamilton systems[J].Chinese Physics B,2012,21(9):094 501.DOI:10.1088/1674-1056/21/9/094501.
[27]Mei F X,Xie J F,Gang T Q.A conformal invariance for generalized Birkhoff equations[J].Acta Mechanica Sinica,2008,24(5):583-585.DOI:10.1007/s10409-008-0176-8.
[28]Cai J L.Conformal invariance of Mei symmetry for the non-holonomic systems of non-Chetaev’s type[J].Nonlinear Dynamics,2011,69(1/2):487-493.
[29]Luo S K,Dai Y,Zhang X T,et al.Fractional conformal invariance method for finding conserved quantities of dynamical systems[J].International Journal of Non-Linear Mechanics,2017,97(12):107-114.
[30]Luo S K,Dai Y,Zhang X T,et al.Basic theory of fractional conformal invariance of Mei symmetry and its applications to physics[J].International Journal of Theoretical Physics,2018,57(4):1 024-1 038.DOI:10.1007/s10773-017-3635-9.
[31]李彦敏.变质量非完整力学系统的共形不变性[J].云南大学学报:自然科学版,2010,32(1):52-57.Li Y M.Conformal invariance for nonholonomic mechanical systems with variable mass[J].Journal of Yunnan University:Natural Sciences Edition,2010,32(1):52-57.
[32]郑世旺.单面完整约束系统Tzénoff方程Mei对称性的共形不变性与守恒量[J].云南大学学报:自然科学版,2018,40(1):74-81.Zheng S W.On conformal invariance and conserved quantity of Mei symmetry for Tzénoff equation in unilateral constraint system[J].Journal of Yunnan University:Natural Sciences Edition,2018,40(1):74-81.
[33]张毅,梅凤翔.基于Riesz分数阶导数的分数阶运动微分方程[J].北京理工大学学报,2012,32(7):766-770.DOI:10.3969/j.issn.1001-0645.2012.07.023.Zhang Y,Mei F X.Fractional differential equations of motion in terms of Riesz fractional derivatives[J].Transactions of Beijing Institute of Technology,2012,32(7):766-770.