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

TL;DR

This paper develops an operator-based framework for classical mechanics in two dimensions, applying it to Kepler and harmonic potentials, deriving classical wave functions, and demonstrating their equivalence via the Levi-Civita map and reparametrization.

Contribution

It introduces an operator formulation of classical mechanics, constructs Hamiltonian operators for specific potentials, and establishes their equivalence using the Levi-Civita map.

Findings

Classical wave functions are finite-region renormalizable.

The operator formulation reproduces classical dynamics.

Kepler and harmonic oscillator models are shown to be equivalent.

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).