The Study Of Pfaff-birkhoff Variational Problem With Time Delay And Its Symmetries

This paper deals with the Pfaff-Birkhoff variational problem with time delay and its symmetries based upon the integer model and the fractional model.Firstly, the Pfaff-Birkhoff principle with time delay is proposed, and the differential equations of motion of Birkhoffian systems with time delay are obtained. According to the invariance of the Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasi-symmetric transformations of systems are defined, and the corresponding criteria of the Noether symmetries are established. The relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for Birkhoffian systems with time delay are established.Furthermore, the generalized Pfaff-Birkhoff principle with time delay is proposed, and the differential equations of motion of generalized Birkhoffian systems with time delay are obtained. The generalized Noether quasi-symmetric transformations of systems are defined, and the corresponding criterion of the Noether symmetries is established. Then the Noether theorem for generalized Birkhoffian systems with time delay is established. Two special cases of generalized Birkhoffian systems with time delay are discussed. The Noether theorems of constrained Birkhoffian systems with time delay and corresponding free Birkhoffian systems with time delay are studied.Then,the fractional Pfaff–Birkhoff principle with time delay in terms of the Riemann-Liouville fractional derivatives is proposed, and the differential equations of motion of fractional Birkhoffian systems with time delay are obtained. Based on the invariance of the fractional Pfaff action with time delay under a group of infinitesimal transformations, the Noether symmetric transformations and the Noether quasi-symmetric transformations of systems are defined, and the corresponding criteria of the Noether symmetries are established. And the relationship between the symmetries and conserved quantities are studied, and the Noether theorems for fractional Birkhoffian systems with time delay are established.Finally, we make a conclusion to this paper, and some remarks on future developments are given.