Two comprehensions on Noether symmetry

This paper presents two ways of comprehending the Noether symmetry for the Lagrange system. One is based on invariance of the Lagrangian, the other is based on invariance of the action. This paper proves that these two comprehensions are different from each other. We give the condition under which the invariance of the Lagrangian can become the invariance of the action, and the condition under which the invariance of the action can become the invariance of the Lagrangian. It is suitable that the Noether symmetry is comprehended as the invariance of the action.