Operator formulation of Classical mechanics: Levi-Civita map and equivalence of central forces in 2-dimensions

Abstract

We study the operator formulation of classical mechanics by explicitly applying it to two central potentials in 2 dimensions. After constructing the classical Hamiltonian operators and corresponding Schr\"odinger like equations, we solve for the corresponding classical wave functions associated with these two potentials, viz; Kepler and harmonic potentials. While satisfying continuity equations, these classical wave functions are shown to be renormalizable only in a finite region of the 2D plane. We also derive the well-known equivalence between these two models within the operator formulation of classical mechanics. This equivalence is shown by relating the Schr\"odinger-like equations and corresponding classical wave functions of these two systems, using the Levi-Civita map and a reparametrizaton of time(Sundman map).