| The time scale,which unify continuous and discrete analysis,is a nonempty closed subset on the field of real numbers.So it also can be set up for the dynamic system,the discrete system and continuous system can be unified.With using the method of time scale,we can extend the theory of classical mechanics to arbitrary time scales.The most important achievement of this paper is unified the two kinds of Mei symmetry theory of mechanical systems,one is in the continuous mechanical systems,the other one is in discrete mechanical systems,as the same time,and presents a method for Mei symmetry of the Lagrange system in conservative mechanical systems and Nielsen system holonomic mechanical systems and its corresponding conserved quantity,which are on time scales.The method develops the basic theory of Mei symmetries and corresponding conserved quantities under the discrete and continuous in the classical dynamical system,unifying both them.First of all,we study the structural equation of the Mei symmetry for the Lagrange equation in conservative mechanical systems on time scales and the three Mei conserved quantities which derived directly from it.Under the infinitesimal transformation of groups,the definition and criterion to Mei symmetry of Lagrange equation on time scale are given,structural equations of Mei symmetry and Mei conserved quantity to Lagrange equation on time scales can also be gotten.Secondly,this paper studies two methods to proving Nielsen equation for holonomic mechanical systems on time scales.One way which combines the JORDAN-HOLDER Theorem with kinetic energy function on time scale.another method to prove the Nielsen equation for holonomic mechanical systems on time scales,with Hamilton principle and Hamilton canonical equation on time scales.Finally,we study the Mei symmetry of the Nielsen equation in the holonomic mechanical system and its Mei conserved quantity on time scales.In this paper,we give the definition and criterion of Mei symmetry of Nielsen equation on time scales,and obtain the structural equation of Mei symmetry of Nielsen equation on time scales and the Mei conserved quantity derived directly from it. |
