Integrating Factor And Conserved Quantity Of The Fractional Action-like Birkhoffian System

In 2005,when El-Nabulsi studied the problem of quantizing the damped harmonic oscillators,a new non-conservative dynamic model was proposed according to the definition of Riemann-Liouville fractional integral.The form of the model is relatively simple and easily calculated,which is called the basic fractional action-like model.Subsequently,in order to further study the fractional variational principle,El-Nabulsi put forward based on the extended exponentially fractional integral and based on the fractional integral extended by periodic law and so on the fractional action-like models.In this paper,the conservation law of Birkhoffian system based on three kinds of the fractional action-like model is studied by the method of integral factor.Research on the conservation theorem of generalized Birkhoffian system is based on El-Nabulsi fractional model.The integral factor of generalized El-Nabulsi-Birkhoff equation is constructed.The necessary condition for the existence of the system conservation quantity is studied.The conservation theorem of is established.The generalized Killing equation used to determine the integral factor is given.Both the non-conservative Hamiltonian and Lagrangian system,based on El-Nabulsi fractional order model,are discussed as special cases,respectively.Research on the conservation theorem of Birkhoffian system is based on fractional integral extended by exponential law.The integral factor of El-Nabulsi-Birkhoff equation based on the extended exponentially fractional integral is constructed.The necessary condition for the existence of the system conservation quantity is studied.The corresponding conservation theorem is established.The generalized Killing equation used to determine the integral factor is given.The non-conservative Hamiltonian system and the non-conservative Lagrangian system based on the extended exponentially fractional integral are discussed as special cases.Research on the conservation theorem of Birkhoffian system is based on fractional integral extended by periodic law.The integral factor of El-Nabulsi-Birkhoff equation based on the fractional integral extended by periodic law is constructed.The necessary condition for the existence of the system conservation quantity is studied.The corresponding conservation theorem is established.The generalized Killing equation used to determine the integral factor is given.The non-conservative Hamiltonian system and the non-conservative Lagrangian system based on the fractional integral extended by periodic law are discussed as special cases.