Lie Symmetries And Conserved Quantities Of Constrained Mechanical Systems On Time Scales

This paper deal with the Lie symmetry and conserved quantity for constrained mechanical systems on time scales,and its symmetries based upon the invariance of the dynamic equations on time scales under the infinitesimal transformations of groups.The Lie theory for non-conservative systems,nonholonomic systems of non-Chetaev's type and Hamiltonian system are given.Constructing a new theory to study Lie symmetry simplifies the complexity of dynamical equations,by the method of time scale theory.It can study the conditions for the formation of Lie symmetry in different dynamical systems on time scales,and the corresponding conserved quantity is established.The study not only simplifies the complexity of dynamic equations,but also extends the development of analytical mechanics.Firstly,we base on the invariance of the dynamic equations on time scales under the infinitesimal transformations of groups,the determining equations of Lie symmetry on time scales are provided by combining the partial differentiation and Taylor's formula.The structure equations of Lie symmetry for non-conservative systems on time scales are established,and form of the Noether conserved quantity is built.Secondly,on the basis of time scales,the dynamic equations of motion for nonholonomic systems of non-Chetaev's type are given.Based on the invariance of differential equation under the infinitesimal transformations,the determining equation is established and the restriction equations for nonholonomic systems of non-Chetaev's type are given.Then,the Lie theorem and conserved quantity for nonholonomic systems of non-Chetaev's type on time scales are built.Finally,based on the principle of Hamilton,the corresponding Hamilton canonical equations are deduced.Based upon the invariance of dynamic equation under the infinitesimal transformations,the determination of the Lie symmetry for the Hamiltonian system on time scales are established.Then,the restriction equations of the Lie symmetry for the Hamiltonian system on time scales are given and form of the Noether conserved quantity are obtained.