Fractional Pfaff-Birkhoff Principle and Birkhoff＇s Equations in Terms of Riesz Fractional Derivatives

The dynamical and physical behavior of a complex system can be more accurately described by using the fractional model. With the successful use of fractional calculus in many areas of science and engineering, it is necessary to extend the classical theories and methods of analytical mechanics to the fractional dynamic system. Birk- hoffian mechanics is a natural generalization of Hamiltonian mechanics, and its core is the PfMf-Birkhoff principle and Birkhoffrs equations. The study on the Birkhoffian mechanics is an important developmental direction of modern analytical mechanics. Here, the fractional Pfaff-Birkhoff variational problem is presented and studied. The definitions of fractional derivatives, the formulae for integration by parts and some other preliminaries are firstly given. Secondly, the fractional Pfaff-Birkhoff principle and the fractional Birkhoff＇s equations in terms of Riesz- Riemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively. Finally, an example is given to illustrate the application of the results.