Noether''s theory of Lagrange systems in discrete case

In this paper, Noether theory of Lagrange systems in discrete case are studied. First, we briefly overview the well-known Noether theory of Lagrange system in the continuous case. Then, we introduce some definitions and notations, such as the operators of discrete translation to the right and the left and the operators of discrete differentiation to the right and the left, and give the conditions for the invariance of the difference functional on the uniform lattice and the non-uniform one, respectively. We also deduce the discrete analog of the Noether-type identity. Finally, the discrete analog of Noether's theorem is presented. An example was discussed to illustrate these results.