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

TL;DR

The paper develops an operator formulation of classical mechanics by constructing a classical wave function whose Schrödinger-like evolution encodes the Hamilton-Jacobi and continuity equations, then applies it to the 2D Kepler problem and the 2D harmonic oscillator. Using the Levi-Civita map and a Sundman-type time reparameterization, it explicitly maps the Hamilton-Jacobi, continuity, and Schrödinger-like equations, as well as the associated wave functions, between these two central-force systems on constant-energy surfaces. The work provides explicit expressions for the classical wave functions, shows that normalization occurs over finite radial intervals, and derives precise energy-constant mappings $E=-2c^2 k'$ and $k=4c^2 E'$ (with $c=1/4$ for a clean wave-function correspondence). Overall, it demonstrates the robustness of the operator approach in reproducing known classical-physics equivalences and suggests avenues toward exploring classical-quantum connections, such as entanglement, within this formalism.

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 Schrödinger 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 Schrödinger-like equations and corresponding classical wave functions of these two systems, using the Levi-Civita map and a reparametrizaton of time(Sundman map).