Noether theorem for non-conservative systems with time delay on time scales

This paper deals with the Noether theorem for non-conservative systems with time delay on time scales. First, the Hamilton principle for the non-conservative system with time delay on time scales is proposed, and Lagrange equations with time delay on time scales are obtained. Second, based on the invariance of the Hamilton action for the non-conservative system with time delay under a group of infinitesimal transformations, the Noether symmetric transformations, the Noether quasi-symmetric transformations and the generalized Noether quasi-symmetric transformations of the system are defined on time scales, respectively, and the criteria of the Noether symmetries are established. Finally, the relationship between the symmetries and the conserved quantities are studied, and the Noether theorems for non-conservative systems with time delay on time scales are established. Some examples are given to illustrate the application of the results.