Noether Theorem For Constraint Mechanical Systems On Time Scales In Event Space

The Noether symmetry and the conserved quantity for mechanical systems not only had a mathematical significance,but also expressed a profound physical law.The Noether symmetry and the conserved quantity for constraint mechanical systems on time scales in event space are studied in this paper.First,the history and the development of Noether theorem were introduced in the configuration space,in the event space and on time scales respectively,as well as the main contents were summarized of this paper.Then,some basic knowledge of the calculus on time scales used in the article are presented,like the forward jump operator,the backward jump operator,the graininess function,the delta derivative and so on.The Noether symmetry and the conserved quantity on time scales in event space are studied.The Lagrange of parameter form on time scales in event space are established.And the Euler-Lagrange equation and the second Euler-Lagrange equation on time scales in event space are established.Based upon the invariance of the Hamilton action on time scales in event space under the infinitesimal transformations of a group,the Noether symmetric relation was obtained and the conserved quantity was obtained by the Noether symmetric relation resulted on time scales in event space.The Noether symmetry and the conserved quantity for a Hamiltonian system on time scales in event space are studied.The Lagrange parametric equations on time scales in event space are given.The generalized momentum and the Hamiltonian on time scales in event space are introduced.The variational problem for a Hamiltonian system on time scales in event space are proposed and established.The Hamiton canonical equations on time scales in event space are obtained.Based upon the invariance of the Hamiltonian action on time scales under the infinitesimal transformations of a group,the definition of the Noether symmetry for a Hamiltonian system on time scales in event space are given.Using the technique of time-re-parameterization,we obtain the Noether theorem for Hamiltonian system on time scales in event space.The Noether symmetry and the conserved quantity for a Birkhoffian system on time scales in event space are studied in this paper.The variational problem for a Birkhoffian system on time scales in event space are proposed and established.Birkhoff equations on time scales in event space are obtained.Based upon the invariance of the Pfaff action on time scales under the infinitesimal transformations of a group,the definition of the Noether symmetry for a Birkhoffian system on time scales in event space are given.Using the technique of time-re-parameterization,we obtain the Noether theorem for Birkhoffian system on time scales in event space.Last,we make a conclusion to this paper and give a forecast to the future.There are three novelties in the paper: first,the parametric equations in the constrained mechanical systems on time scales in event space are established.Second,The Noether symmetry and the conserved quantity for constraint mechanical systems on time scales in event space are obtained.Last,we extend the theories,we prove the Noether theorem in configuration space,in event space and on time scales are the special case.