Equations of motion, Noncommutativity and Quantization

TL;DR

This paper explores the connection between equations of motion, noncommutative symplectic structures, and quantization, establishing conditions for consistency in both commutative and noncommutative cases.

Contribution

It derives the consistency conditions for noncommutative symplectic structures related to equations of motion, extending classical Helmholtz conditions.

Findings

Established conditions for noncommutative symplectic forms

Linked equations of motion with Poisson brackets in noncommutative settings

Extended Helmholtz integrability to noncommutative cases

Abstract

We study the relation between a given set of equations of motion in configuration space and a Poisson bracket. A Poisson structure is consistent with the equations of motion if the symplectic form satisfy some consistency conditions. When the symplectic structure is commutative these conditions are the Helmholtz integrability equations for the nonrestricted inverse problem of the calculus of variations. We have found the corresponding consistency conditions for the symplectic noncommutative case.