Features of free particles system motion in noncommutative phase space

TL;DR

This paper investigates how noncommutativity in phase space affects the motion of free particles, revealing mass-dependent trajectories and the conditions under which total momentum remains conserved.

Contribution

It demonstrates the impact of phase space noncommutativity on free particle dynamics and identifies conditions for conserved total momentum in such systems.

Findings

Particles with the same initial velocities diverge due to noncommutativity.

Mass influences particle trajectories and velocities in noncommutative space.

Total momentum is not conserved unless noncommutativity parameters depend on mass.

Abstract

Influence of noncommutativity on the motion of composite system is studied in noncommutative phase space of canonical type. A system composed by $N$ free particles is examined. We show that because of momentum noncommutativity free particles of different masses with the same velocities at the initial moment of time do not move together. The trajectory and the velocity of free particle in noncommutative phase space depend on its mass. So, a system of the free particles flies away. Also, it is shown that the total momentum defined in the traditional way is not integral of motion in a space with noncommutativity of coordinates and noncommutativity of momenta. We find that in the case when parameters of noncommutativity corresponding to a particle are determined by its mass the trajectory and velocity of free particle are independent of the mass, also the total momenta as integrals of motion can be introduced in noncommutative phase space.