Mechanical similarity as a generalization of scale symmetry

In this paper we study the symmetry known (Landau and Lifshits 1976 Course of Theoretical Physics Vol.1: Mechanics (Oxford: Pergamon } as mechanical similarity (LMS) and present for any monomial potential. We analyze it in the framework of the Koopman-von Neumann formulation of classical mechanics and prove that in this framework the LMS can be given a canonical implementation. We also show that the LMS is a generalization of the scale symmetry which is present only for the inverse square potential. Finally we study the main obstructions which one encounters in implementing the LMS at the quantum mechanical level.