基于按周期律拓展的El-Nabulsi分数阶模型的Birkhoff系统的积分因子方法
|
摘要
为了进一步研究积分因子方法理论在分数阶模型中的应用,将积分因子方法应用于基于按周期律拓展的Birkhoff系统,建立了寻找基于按周期律拓展的Birkhoff系统守恒量的一种新方法。首先,给出基于按周期律拓展的分数阶El-Nabulsi-Birkhoff方程,并定义出方程的积分因子;其次,详细地研究了守恒量存在的必要条件,同时找出积分因子与守恒量之间的关系,得出了广义Killing方程,建立相应的守恒定理;最后,通过举例来说明结果的应用。
In order to further study the application of integrating factor method in fractional order model,we applied it to the Birkhoff system extended by periodic law and proposed a new method for finding the conserved quantities of the Birkhoff system. Firstly,the fractional order El-Nabulsi-Birkhoff equation extended by periodic law was given,and the integrating factor of equation was defined. Secondly,the necessary conditions for the existence of conserved quantities were studied,and the relationship between the integral factor and the conserved quantity was explored. A generalized Killing equation was obtained,and the corresponding conservation theorem was established. Finally,an example was given to illustrate the application of the results.
In order to further study the application of integrating factor method in fractional order model,we applied it to the Birkhoff system extended by periodic law and proposed a new method for finding the conserved quantities of the Birkhoff system. Firstly,the fractional order El-Nabulsi-Birkhoff equation extended by periodic law was given,and the integrating factor of equation was defined. Secondly,the necessary conditions for the existence of conserved quantities were studied,and the relationship between the integral factor and the conserved quantity was explored. A generalized Killing equation was obtained,and the corresponding conservation theorem was established. Finally,an example was given to illustrate the application of the results.
引文
[1]OLDHAM K B,SPANIER J.The Fractional Calculus[M].San Diego:Academic Press,1974.
[2]PODLUBNY I.Fractional Differential Equation[M].San Diego:Academic Press,1999.
[3]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and Applications of Fractional Differential Equations[M].Amsterdam:Elsevier B V,2006.
[4]RIEWE F.Nonconservative Lagrangian and Hamiltonian mechanics[J].Physical Review E,1996,53(2):1890-1899.
[5]AGRAWAl O P.Formulation of Euler-Lagrange equations for fractional variational problems[J].Journal of Mathematical Analysis&Applications,2002,272(1):368-379.
[6]JUMARIE G.Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function[J].Journal of Applied Mathematics&Computing,2007,23(1/2):215-228.
[7]ATANACKOVIC,TEODOR M,KONJIK S,et al.Variational problems with fractional derivatives:invariance conditions and Noether's theorem[J].Nonlinear Analysis Theory Methods&Applications,2009,71(5/6):1504-1517.
[8]AHMAD-RAMi E N.A fractional approach of nonconservative Lagrangian dynamics[J].Fizika A,2005,14(4):289-298.
[9]RAMI E N 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.
[10]丁金凤,张毅.基于El-Nabulsi动力学模型的Birkhoff力学[J].苏州科技学院学报(自然科学版),2014,31(1):24-28.
[11]DJUKIC 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.
[12]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.
[13]QIAO Y F,MENG J.Existential theorem of conserved quantities and its inverse for the dynamics of nonholonomic relativistic systems[J].中国物理B:英文版,2002,11(9):859-863.
[14]QIAO Y F,ZHANG Y L.Integrating factors and conservation theorem for holonomic nonconservative dynamical systems in generalized classical mechanics[J].Chinese Physics B,2002,11(10):988-992.
[15]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.
[16]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.
[17]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.
[2]PODLUBNY I.Fractional Differential Equation[M].San Diego:Academic Press,1999.
[3]KILBAS A A,SRIVASTAVA H M,TRUJILLO J J.Theory and Applications of Fractional Differential Equations[M].Amsterdam:Elsevier B V,2006.
[4]RIEWE F.Nonconservative Lagrangian and Hamiltonian mechanics[J].Physical Review E,1996,53(2):1890-1899.
[5]AGRAWAl O P.Formulation of Euler-Lagrange equations for fractional variational problems[J].Journal of Mathematical Analysis&Applications,2002,272(1):368-379.
[6]JUMARIE G.Fractional Hamilton-Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function[J].Journal of Applied Mathematics&Computing,2007,23(1/2):215-228.
[7]ATANACKOVIC,TEODOR M,KONJIK S,et al.Variational problems with fractional derivatives:invariance conditions and Noether's theorem[J].Nonlinear Analysis Theory Methods&Applications,2009,71(5/6):1504-1517.
[8]AHMAD-RAMi E N.A fractional approach of nonconservative Lagrangian dynamics[J].Fizika A,2005,14(4):289-298.
[9]RAMI E N 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.
[10]丁金凤,张毅.基于El-Nabulsi动力学模型的Birkhoff力学[J].苏州科技学院学报(自然科学版),2014,31(1):24-28.
[11]DJUKIC 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.
[12]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.
[13]QIAO Y F,MENG J.Existential theorem of conserved quantities and its inverse for the dynamics of nonholonomic relativistic systems[J].中国物理B:英文版,2002,11(9):859-863.
[14]QIAO Y F,ZHANG Y L.Integrating factors and conservation theorem for holonomic nonconservative dynamical systems in generalized classical mechanics[J].Chinese Physics B,2002,11(10):988-992.
[15]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.
[16]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.
[17]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.