  In this work we have shown precisely that the curvature of a 2-sphere
introduces quantum features in the system through the introduction of the
noncommutative (NC) parameter that appeared naturally via equations of motion.
To obtain this result we used the fact that quantum mechanics can be understood
as a NC symplectic geometry, which generalized the standard description of
classical mechanics as a symplectic geometry. In this work, we have also
analyzed the dynamics of the model of a free particle over a 2-sphere in a NC
phase-space. Besides, we have shown the solution of the equations of motion
allows one to show the equivalence between the movement of the particle
physical degrees of freedom upon a 2-sphere and the one described by a central
field. We have considered the effective force felt by the particle as being
caused by the curvature of the space. We have analyzed the NC Poisson algebra
of classical observables in order to obtain the NC corrections to Newton's
second law. We have demonstrated precisely that the curvature of the space
acted as an effective potential for a free particle in a flat phase-space.
Besides, through NC coherent states quantization we have obtained the Green
function of the theory. The result have confirmed that we have an UV cutoff for
large momenta in the NC kernel.
  We have also discussed the relation between affine connection and Dirac
brackets, as they describe the proper evolution of the model over the surface
of constraints in the Lagrangian and Hamiltonian formalisms, respectively. As
an application, we have treated the so-called \textit{Zitterbewegung} of the
Dirac electron. Since it is assumed to be an observable effect, then we have
traced its physical origin by assuming that the electron has an internal
structure.
