Vertical extension of Noether theorem for scaling symmetries

The aim of this paper is to present a new approach construct constants motion associated with scaling symmetries dynamical systems. Scaling maps could be the equations but not its Lagrangian action. We have constructed Noether inspired theorem in vertical extended space that can used obtain for these symmetries. obtained as particular case our construction. To illustrate how procedure works, we two interesting examples, (a) Schwarzian Mechanics based on derivative operator and (b) Korteweg–de Vries nonlinear partial differential equation context asymptotic dynamics General Relativity $$\hbox {AdS}_3$$ . also study inverse show identify generator transformation, it works constant are able construct. find an contribution symmetry fact Finally, contrasted results recent analysis previous attempts beautiful laws.