HOJMAN CONSERVED QUANTITY FOR TIME SCALES LAGRANGE SYSTEMS

By using symmetry and conservation laws, we can simplify dynamical problem and even obtain the exact solution of mechanical system, and better understand the dynamical behavior of system. Time scales analysis unifies and extends the continuous and discrete dynamics models to the time scales framework, which not only avoids repeated studies but also reveals the differences and connections between them. Therefore, it is necessary to explore new conservation laws in the framework of time scale through symmetry. Firstly, the Lagrange equations on time scales are established, and two important relations of time scales Lagrange system are derived by using the properties of time scales calculus. Secondly, according to the invariance of differential equation under the one-parameter Lie group of transformations, the definition of Lie symmetry on time scales and its determining equation are established. Thirdly, the Lie symmetry theorem on time scales is established and proved by using the above relations, and the new conservation laws of time scales Lagrange system are obtained. When the time scale is taken to the set of real numbers, the conservation laws degenerate to the famous Hojman conserved quantity. Finally, a two-degree-of-freedom time scales Lagrange system is investigated, and its Hojman conserved quantities are obtained in three different time scales, and the correctness of the theorem we obtained is verified by numerical calculation.