The Fractional Pfaff-birkhoff Variational Problem And Its Symmetries Based On El-nabulsi Model

In order to model the nonconservative system El-Nabulsi put forward three kindsof fractional action-like variational approaches, the fractional variational problembased on the definition of the Riemann-Liouville fractional integral, the fractionalvariational problem based on exponentially extended fractional integral, and thefractional variational problem based on the fractional integral extended by periodic law.The above-mentioned approaches are called El-Nabulsi dynamical models. In thispaper we will extend the El-Nabulsi dynamical models to Birkhoff system, establishthe corresponding El-Nabulsi-Pfaff-Birkhoff principles and the El-Nabulsi-Birkhoffequations, study the Noether theorey of the three models. The main contents of thepaper as follows:1. Noether symmetries and the conserved quantities for thevariational problem of generalized Birkhoffian system based on the definition of theRiemann-Liouville fractional integral are studied.2. Noether symmetries and theconserved quantities for the variational problem of Birkhoffian system based onexponentially extended fractional integral are studied.3. Noether symmetries and theconserved quantities for the variational problem of Birkhoffian system based on thefractional integral extended by periodic law are studied.1. Noether’s theorem of a generalized Birkhoffian system is proposed and studiedwhich is based on the definition of the Riemann-Liouville fractional integral. Firstly,the generalized El-Nabulsi-Pfaff-Birkhoff principle based on the definition of theRiemann-Liouville fractional integral is established, then the correspondinggeneralized El-Nabulsi-Birkhoff equations are derived. Secondly, the definitions andcriteria of the system are given, which are based on the invariance of El-Nabulsi-Pfaffaction under the infinitesimal transformations of group. Then the Noether theorem forthe variational problem of generalized Birkhoffian system is established. Finally, Thenonconservative Hamilton and Lagrange system based on Riemann-Liouvillefractional integral of El-Nabulsi model, as well as the classical Birkhoff system are thespecial ensamples of the generalized Birkhoffian system based on the definition of theRiemann-Liouville fractional integral. Illustrate the special ensamples to discuss theapplication of the results.2. Noether symmetries and the conserved quantities for the variational problem ofBirkhoffian system based on extended exponentially fractional integral are presentedand studied. Firstly, the El-Nabulsi-Pfaff variational problem based on extendedexponentially fractional integral is presented, then the correspondingEl-Nabulsi-Birkhoff equations are derived. Secondly, the definitions and criteria of thesystem are given, which are based on the invariance of El-Nabulsi-Pfaff action under the infinitesimal transformations of group. Then the Noether theorem for thevariational problem of Birkhoffian system is established. The nonconservativeHamilton and Lagrange system based on Riemann-Liouville fractional integral ofEl-Nabulsi model, as well as the classical Birkhoff system are the special ensamples ofthe generalized Birkhoffian system based on the definition of the Riemann-Liouvillefractional integral. Illustrate the special ensamples to discuss the application of theresults.3. Noether symmetries and the conserved quantities for the variational problem ofBirkhoffian system based on the fractional integral extended by periodic law arepresented and studied. Firstly, the El-Nabulsi-Pfaff variational problem based on thefractional integral extended by periodic law is presented, then the correspondingEl-Nabulsi-Birkhoff equations are derived. Secondly, the definitions and criteria of thesystem are given, which are based on the invariance of El-Nabulsi-Pfaff action underthe infinitesimal transformations of group. Then the Noether theorem for thevariational problem of Birkhoffian system is established. The nonconservativeHamilton and Lagrange system based on Riemann-Liouville fractional integral ofEl-Nabulsi model, as well as the classical Birkhoff system are the special ensamples ofthe generalized Birkhoffian system based on the definition of the Riemann-Liouvillefractional integral. Illustrate the special ensamples to discuss the application of theresults.