Classical Mechanics on Noncommutative Space with Lie-algebraic Structure

TL;DR

This paper explores classical particle motion on Lie-algebraic noncommutative spaces, revealing new trajectories influenced by complex extra forces due to noncommutativity, extending previous models and providing novel insights into noncommutative classical mechanics.

Contribution

The study introduces two new algebraic structures for noncommutative spaces and analyzes their impact on particle trajectories using Hamiltonian formalism, expanding the understanding of classical mechanics in noncommutative geometry.

Findings

Derived new algebraic structures for noncommutative spaces.

Identified complex extra forces affecting particle trajectories.

Extended previous models to include novel noncommutative effects.

Abstract

We investigate the kinetics of a nonrelativistic particle interacting with a constant external force on a Lie-algebraic noncommutative space. The structure constants of a Lie algebra, also called noncommutative parameters, are constrained in general due to some algebraic properties, such as the antisymmetry and Jacobi identity. Through solving the constraint equations the structure constants satisfy, we obtain two new sorts of algebraic structures, each of which corresponds to one type of noncommutative spaces. Based on such types of noncommutative spaces as the starting point, we analyze the classical motion of the particle interacting with a constant external force by means of the Hamiltonian formalism on a Poisson manifold. Our results {\em not only} include that of a recent work as our special cases, {\em but also} provide new trajectories of motion governed mainly by marvelous extra forces. The extra forces with the unimaginable $t\dot{x}$-, $\dot{(xx)}$-, and $\ddot{(xx)}$-dependence besides with the usual $t$-, $x$-, and $\dot{x}$-dependence, originating from a variety of noncommutativity between different spatial coordinates and between spatial coordinates and momenta as well, deform greatly the particle's ordinary trajectories we are quite familiar with on the Euclidean (commutative) space.