Noncommutative Lagrange Mechanics

TL;DR

This paper introduces a geometric approach to incorporate noncommutativity into classical mechanics without relying on quantum-inspired formalisms, analyzing its effects on equations of motion and system dynamics.

Contribution

It presents a novel geometric formulation of noncommutative classical mechanics, emphasizing internal configuration space structure and deriving modified equations of motion.

Findings

Noncommutativity modeled as an internal geometric structure.

Modified equations include a quadratic Lorentz-like force.

Noncommutative effects mediated by background interactions.

Abstract

It is proposed how to impose a general type of ''noncommutativity'' within classical mechanics from first principles. Formulation is performed in completely alternative way, i.e. without any resort to fuzzy and/or star product philosophy, which are extensively applied within noncommutative quantum theories. Newton-Lagrange noncommutative equations of motion are formulated and their properties are analyzed from the pure geometrical point of view. It is argued that the dynamical quintessence of the system consists in its kinetic energy (Riemannian metric) specifying Riemann-Levi-Civita connection and thus the inertia geodesics of the free motion. Throughout the paper, ''noncommutativity'' is considered as an internal geometric structure of the configuration space, which can not be ''observed'' per se. Manifestation of the noncommutative phenomena is mediated by the interaction of the system with noncommutative background under the consideration. The simplest model of the interaction (minimal coupling) is proposed and it is shown that guiding affine connection is modified by the quadratic analog of the Lorentz electromagnetic force (contortion term).