Noether theorem for vakonomic dynamics on time scales

This research is dedicated to studying the Noether symmetry and conservation laws of vakonomic dynamics on an arbitrary time scale. First, the variational principle of vakonomic dynamics on time scales is established, from which the equations of motion are derived. Second, the Noether symmetry of such systems is defined, and the corresponding criterion, referred to as the Noether identity, is provided. Third, the Noether theorem for vakonomic dynamics on time scales is formulated and proved, leading to the derivation of the Noether conserved quantity. Under appropriate conditions, the results reduce to the classical Noether theorem for vakonomic dynamics. Finally, a specific example is presented to demonstrate the application of the theorem and to verify its validity.