Galilean-Invariant (2+1)-Dimensional Models with a Chern-Simons-Like Term and D=2 Noncommutative Geometry

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 such a 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 a Lagrangian as well as Hamiltonian framework and it is interpreted as describing the superposition of the free motion in noncommutative D=2 space and the ``internal'' oscillator modes. Further we add a suitably chosen class of velocity-dependent two-particle interactions, which describe local potentials in the D=2 noncommutative space. We treat, in detail, the particular case of an oscillator and describe its quantization. Finally we show that the indefinite metric due to the third time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposition of a subsidiary condition.