Herglotz Variational Problem And Symmetry For Non-conservative Systems With Time Delay

In mechanics,the variational principle refers to the problem of calculating the functional extremum with the variational method.However,the classical variational principle is difficult to describe the dynamic motion process of dissipative system and non-conservative system.Gustav Herglotz proposed Herglotz generalized variational principle,which can not only describe the dynamic equation of non-conservative system,but also describe the conservative dynamical system under the classical variational principle.In this paper,based on the fractional model,we study the Herglotz type adiabatic invariants of Birkhoffian system,and considering the time delay effect of the non-conservative system,the Herglotz type Noether theorem of the system is studied.Noether's theorem for fractional Lagrangian system of Herglotz type with time delay is studied.Firstly,the motion equation of the system can be obtained by Herglotz variational principle.Secondly,two basic formulas of non-isochronous variation of Hamilton-Herglotz action are established.Finally,the Herglotz type Noether theorem of the system is proved.Noether's theorem for fractional Hamiltonian system of Herglotz type with time delay is studied.Firstly,the canonical equation of the system can be obtained by Herglotz variational principle.Secondly,two basic formulas of non-isochronous variation of Hamilton-Herglotz action are established.Finally,the Herglotz type Noether theorem of the system is proved.Noether's theorem for fractional Birkhoffian system of Herglotz type with time delay is studied.Firstly,the motion equation of the system can be obtained by Herglotz variational principle.Secondly,two basic formulas of non-isochronous variation of Pfaff-Herglotz action are established.Finally,the Herglotz type Noether theorem of the system is proved.Noether's theorem for nonholonomic system of Herglotz type with time delay is studied.Firstly,the motion equation of the system can be obtained by Herglotz variational principle.Secondly,two basic formulas of non-isochronous variation of Herglotz action are established.Finally,the Herglotz type Noether theorem of the system is proved.The conserved quantities and adiabatic invariants of Herglotz fractional Birkhoffian system are studied.The fractional Pfaff-Herglotz action is proposed,and the differential equation of the system is obtained,the exact invariants of Herglotz type of the system are proved.Furthermore,the adiabatic invariants of Herglotz type are derived when the system is subjected to small perturbation.