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

作　　者：周燕 张毅 

机构地区：[1]College of Mathematics and Physics,Suzhou University of Science and Technology [2]College of Civil Engineering,Suzhou University of Science and Technology

出　　处：《Transactions of Nanjing University of Aeronautics and Astronautics》2014年第1期63-69,共7页南京航空航天大学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China(10972151,11272227);the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province(CXZZ11_0949);the Innovation Program for Postgraduate of Suzhou University of Science and Technology(SKCX11S_050)

摘　　要：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.Birkhoffian mechanics is a natural generalization of Hamiltonian mechanics,and its core is the Pfaff-Birkhoff principle and Birkhoff′s 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 RieszRiemann-Liouville fractional derivatives and Riesz-Caputo fractional derivatives are presented respectively.Finally,an example is given to illustrate the application of the results.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.

关 键 词：fractional derivative fractional Pfaff-Birkhoff principle fractional Birkhoff′s equation transversality condition 

分 类 号：O316[理学—一般力学与力学基础]