In the general Noether Theorem's proof, it is required that the action before transformation is equal to the action after transformation: I = I'. Who can tell me why this condition has to be used. In my opinion, we can obtain the same form of the motion equation after the transformation only if the Lagrangian density has the same functional form. We don't have to use this scale-invariance condition. If we don't use this condition, can we still obtain the same form of the equation? PS:Which book gives a clear proof of the Noether Theorem?