Noether Symmetry and Conserved Quantities of Fractional Birkhoffian System in Terms of Herglotz Variational Problem**Supported by the National Natural Science Foundation of China under Grant Nos. 11272227 and 11572212, the Innovation Program for Postgraduate in Higher Education Institutions of Jiangsu Province (KYZZ16 0479), and the Innovation Program for Postgraduate of Suzhou University of Science and Technology (SKCX16 058)

Abstract

The aim of this paper is to study the Herglotz variational principle of the fractional Birkhoffian system and its Noether symmetry and conserved quantities. First, the fractional Pfaff-Herglotz action and the fractional Pfaff-Herglotz principle are presented. Second, based on different definitions of fractional derivatives, four kinds of fractional Birkhoff’s equations in terms of the Herglotz variational principle are established. Further, the definition and criterion of Noether symmetry of the fractional Birkhoffian system in terms of the Herglotz variational problem are given. According to the relationship between the symmetry and the conserved quantities, the Noether’s theorems within four different fractional derivatives are derived, which can reduce to the Noether’s theorem of the Birkhoffian system in terms of the Herglotz variational principle under the classical conditions. As applications of the Noether’s t heorems of the fractional Birkhoffian system in terms of the Herglotz variational principle, an example is given at the end of this paper.