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 nece...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.展开更多
This paper presents fractional generalized canonical transformations for fractional Birkhoffian systems within Caputo derivatives.Firstly,based on fractional Pfaff-Birkhoff principle within Caputo derivatives,fraction...This paper presents fractional generalized canonical transformations for fractional Birkhoffian systems within Caputo derivatives.Firstly,based on fractional Pfaff-Birkhoff principle within Caputo derivatives,fractional Birkhoff’s equations are derived and the basic identity of constructing generalized canonical transformations is proposed.Secondly,according to the fact that the generating functions contain new and old variables,four kinds of generating functions of the fractional Birkhoffian system are proposed,and four basic forms of fractional generalized canonical transformations are deduced.Then,fractional canonical transformations for fractional Hamiltonian system are given.Some interesting examples are finally listed.展开更多
In order to study discrete fractional Birkhoff equations for Birkhoffian systems,the method of isochronous variational principle is used in this paper. Discrete fractional Pfaff-Birkhoff principle in terms of time sca...In order to study discrete fractional Birkhoff equations for Birkhoffian systems,the method of isochronous variational principle is used in this paper. Discrete fractional Pfaff-Birkhoff principle in terms of time scales is presented. Discrete fractional Birkhoff equations with left and right discrete operators of Riemann-Liouville type are established and some special cases including classical discrete Birkhoff equations,discrete fractional Hamilton equations and discrete fractional Lagrange equations are discussed. Finally,an example is devoted to illustrate the results.展开更多
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)
文摘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.
基金supported by the National Natural Science Foundations of China(Nos.11972241,11572212,11272227)the Natural Science Foundation of Jiangsu Province(No.BK20191454)。
文摘This paper presents fractional generalized canonical transformations for fractional Birkhoffian systems within Caputo derivatives.Firstly,based on fractional Pfaff-Birkhoff principle within Caputo derivatives,fractional Birkhoff’s equations are derived and the basic identity of constructing generalized canonical transformations is proposed.Secondly,according to the fact that the generating functions contain new and old variables,four kinds of generating functions of the fractional Birkhoffian system are proposed,and four basic forms of fractional generalized canonical transformations are deduced.Then,fractional canonical transformations for fractional Hamiltonian system are given.Some interesting examples are finally listed.
基金National Natural Science Foundations of China(Nos.11272227,11572212)the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province,China(No.KYLX15_0405)
文摘In order to study discrete fractional Birkhoff equations for Birkhoffian systems,the method of isochronous variational principle is used in this paper. Discrete fractional Pfaff-Birkhoff principle in terms of time scales is presented. Discrete fractional Birkhoff equations with left and right discrete operators of Riemann-Liouville type are established and some special cases including classical discrete Birkhoff equations,discrete fractional Hamilton equations and discrete fractional Lagrange equations are discussed. Finally,an example is devoted to illustrate the results.
