The Symmetries of Equivalent Lagrangian Systems and Constants of Motion

In this paper mathematical structure of time-dependent lagrangian systems and their symmetries are extended and the explicit relation between constants of motion and infinitesimal symmetries of time-dependent lagrangian systems are considered. starting point is time-independent lagrangian systems ,then we extend mathematical concepts of these systems such as equivalent lagrangian systems to the case of time-dependent lagrangian systems. also some new theorems and corollaries will be proved. finally we make a 1-1 correspondence between the symmetries of equivalent time-dependent lagrangian systems and constants of motion by the new geometric concept of galilean space-time.