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 analyse 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 and a few other potentials. Finally, we study the main obstructions which one encounters in implementing the LMS at the quantum-mechanical level