Newton equation for canonical, Lie-algebraic and quadratic deformation of classical space

TL;DR

This paper investigates how different types of space-time deformations affect particle motion under external forces, revealing that certain noncommutative geometries introduce additional accelerations and velocity-dependent forces.

Contribution

It provides a comparative analysis of particle dynamics in canonical, Lie-algebraic, and quadratic deformed space-times, highlighting the effects of noncommutativity on classical equations of motion.

Findings

Canonical deformation does not alter particle dynamics.

Lie-algebraic noncommutativity introduces additional acceleration.

Quadratic deformation results in velocity and position-dependent forces.

Abstract

The Newton equation describing the particle motion in constant external field force on canonical, Lie-algebraic and quadratic space-time is investigated. We show that for canonical deformation of space-time the dynamical effects are absent, while in the case of Lie-algebraic noncommutativity, when spatial coordinates commute to the time variable, the additional acceleration of particle is generated. We also indicate, that in the case of spatial coordinates commuting in Lie-algebraic way, as well as for quadratic deformation, there appear additional velocity and position-dependent forces.