  In this work we have investigated some properties of classical phase-space
with symplectic structures consistent, at the classical level, with two
noncommutative (NC) algebras: the Doplicher-Fredenhagen-Roberts algebraic
relations and the NC approach which uses an extended Hilbert space with
rotational symmetry. This extended Hilbert space includes the operators
$\theta^{ij}$ and their conjugate momentum $\pi_{ij}$ operators. In this
scenario, the equations of motion for all extended phase-space coordinates with
their corresponding solutions were determined and a rotational invariant NC
Newton's second law was written. As an application, we treated a NC harmonic
oscillator constructed in this extended Hilbert space. We have showed precisely
that its solution is still periodic if and only if the ratio between the
frequencies of oscillation is a rational number. We investigated, analytically
and numerically, the solutions of this NC oscillator in a two-dimensional
phase-space. The result led us to conclude that noncommutativity induces a
stable perturbation into the commutative standard oscillator and that the
rotational symmetry is not broken. Besides, we have demonstrated through the
equations of motion that a zero momentum $\pi_{ij}$ originated a constant NC
parameter, namely, $\theta^{ij}=const.$, which changes the original variable
characteristic of $\theta^{ij}$ and reduces the phase-space of the system. This
result shows that the momentum $\pi_{ij}$ is relevant and cannot be neglected
when we have that $\theta^{ij}$ is a coordinate of the system.
