Operator Formulation of Classical Mechanics

Slobodan Prvanović

Jack Cohn

An operator formulation of classical mechanics is given. Rather than being concerned with Lie algebraic or abstract Hilbert space properties of classical mechanics, the present formulation is directly concerned with wave packets in configuration space and is thus more similar to that of conventional quantum theory than other extant operator formulations of classical mechanics. Among other things, the following items are discussed for classical systems: Ehrenfest’s theorem, the uncertainty principle, dissipative systems, and field theory.

Jaroslav Zamastil

Paul Richard Halmos, John von Neumann

Philip M. Campbell

A novel formulation of classical dynamics in terms of differential operators is presented. The basis for this formulation is the direct application of the unfolding theorem for arbitrary times. The solution of a mechanical problem is given by the operator method as a Taylor series about the initial point. The classical operator formalism has similarities with quantum mechanics. It is shown that classical scattering can be described in terms of a classical S-matrix operator. Examples are given.

Nikhil Kasyap Puranam, E. Harikumar

We study the operator formulation of classical mechanics by explicitly applying it to two central potentials in two 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 reparametrization of time (Sundman map).