Noether symmetries and conserved quantities for fractional Birkhoffian systems

This paper presents the variational problems for fractional Birkhoffian systems, and its corresponding symmetries and conserved quantities are further studied. First, the fractional Pfaff variational problems in terms of Riemann-Liouville fractional derivatives are proposed, and the fractional Pfaff-Birkhoff-d'Alembert principle is established. Thus, the fractional Birkhoff's equations are derived. Second, we derive two basic formulae for the variation of Pfaff action and give the definitions and criteria of fractional Noether symmetries and quasi-symmetries. Third, we establish the Noether theorems of fractional Birkhoffian systems which reveal the relationship between fractional symmetries and fractional conserved quantities. Some special cases are discussed, showing the universality of our results. Two examples are analyzed to illustrate the applications of the results in detail.