Noether's theorem for a time-delayed Birkhoffian system of Herglotz type

This paper focuses on studying the Noether symmetries and the conserved quantities for a time-delayed Birkhoffian system of Herglotz type. The generalized Herglotz variational principle is extended to a time-delayed Birkhoffian system, and Birkhoff's equations of Herglotz type with time delay are established. The definition and the criterion of Noether symmetric transformation for the time-delayed Birkhoffian system of Herglotz type are given, which are based upon the formulae of non-isochronal variation of the Pfaff-Herglotz action deduced in this paper. Noether's theorem for the time-delayed Birkhoffian system of Herglotz type is proved. The theorem reveals the inherent relation between the Noether symmetries and the conserved quantities of the system, and contains Noether's theorem for the time-delayed Birkhoffian system based on the classical variational problem and Noether's theorem for the time-delayed Hamiltonian system of Herglotz type as its specials. At the end of the paper, an example is given to illustrate the application of the results.