Herglotz Variational Principles Of Constrained Mechanical Systems And Their Noether Symmetries And Conserved Quantities

The Herglotz variational principle is a generalized variational principle.The functional of the Herglotz variational principle is defined by a differential equation,whose extrema are sought.The Herglotz variational principle not only can describe all dynamic processes which can be described by the classical variational principle,but also can give variational descriptions of nonconservative systems and dissipative systems for which the classical variational principle can not be applied.Therefore,by use of the Herglotz variational principle,we can systematically deal with the problems of conservative systems and nonconservative systems.In this paper,based on the Herglotz variational principles,Noether's theorems and inverse theorems for the nonconservative Lagrangian system and the Birkhoffian system in event space are established,respectively.Further,Noether's theorems of Herglotz type on time scales and on fractional order models are investigated.Firstly,according to the Herglotz variational principle,the differential equation of motion for the nonconservative Lagrangian system is deduced,the definition and criterion of the Noether symmetry transformation of Herglotz type are given,and the Noether's theorem and inverse theorem of Herglotz type for the nonconservative Lagrangian system are obtained.Secondly,the Herglotz variational principle for the Birkhoffian system in event space is given,its parametric equations of this system are deduced,the corresponding definition and criterion of Noether symmetry transformation of Herglotz type are provided,and Noether's theorem and inverse theorem of Herglotz type for the Birkhoffian system in event space are obtained.Moreover,the Herglotz variational principles and Noether's theorems for the nonconservative Lagrangian system and the nonconservative Hamiltonian system on time scales are studied.The dynamic equations of Herglotz type on time scales are deduced,the definitions of the Noether symmetry of Herglotz type are given and the Noether identities are derived,and Noether's theorems of Herglotz type on time scales are established.Finally,the Herglotz variational principles and Noether's theorems for the fractional nonconservative Hamiltonian system and the fractional Birkhoffian system are studied.The fractional Herglotz variational principles are given,the differential equations of motion of Herglotz type are deduced,and the corresponding Noether's theorems are obtained on account of the definitions of the Noether symmetry for the fractional Herglotz variational principles.