Noether's theorem for fractional Herglotz variational principle in phase space

The aim of this paper is to bring together two approaches, Heglotz variational principle and fractional calculus, to deal with non-conservative systems in phase space. Namely, we study the functional of Herglotz type whose extremum is sought, by the differential equation that involves Caputo fractional derivatives in phase space. Firstly, Herglotz variational principle under fractional Hamilton action in phase space is presented, and its Hamilton canonical equations are derived. Secondly, two basic formulae for the variation of the fractional Hamilton-Herglotz action in phase space are obtained. Furthermore, the definition and the criterion of Noether symmetry for fractional Herglotz variational principle are given, and the corresponding Noether's theorem is established. Under appropriate conditions, the Noether's theorem can reduce to the classical one of Herglotz type in phase space. Finally, two examples are given to illustrate the application of the results. (c) 2018 Elsevier Ltd. All rights reserved.