  We consider a new D=2 nonrelativistic classical mechanics model providing via
the Noether theorem the (2+1)-Galilean symmetry algebra with two central
charges: mass m and the coupling constant k of a Chern-Simons-like term. In
this way we provide the dynamical interpretation of the second central charge
of the (2+1)-dimensional Galilean algebra. We discuss also the interpretation
of k as describing the noncommutativity of D=2 space coordinates. The model is
quantized in two ways: using the Ostrogradski-Dirac formalism for higher order
Lagrangians with constraints and the Faddeev-Jackiw method which describes
constrained systems and produces nonstandard symplectic structures. We show
that our model describes the superposition of a free motion in noncommutative
D=2 space as well as the "internal" oscillator modes. We add a suitably chosen
class of velocity-dependent two-particle interactions, which is descrobed by
local potentials in D=2 noncommutative space. We treat, in detail, the
particular case of a harmonic oscillator and describe its quantization. It
appears that the indefinite metric due to the third order time derivative term
in the field equations, even in the presence of interactions, can be eliminated
by the imposition of a subsidiary condition.
