GALILEAN-INVARIANT (2-DIMENSIONAL MODELS WITH A CHERN-SIMONS-LIKE TERM AND D=2 NONCOMMUTATIVE GEOMETRY(1))

We consider a new D = 2 nonrelativistic classical mechanics model prov iding via the Noether theorem the (2 + 1)-Galilean symmetry algebra wi th 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 algeb ra. We discuss also the interpretation of k as describing the noncommu tativity of D = 2 space coordinates. The model is quantized in two way s: using the Ostrogradski-Dirac formalism for higher order Lagrangians with constraints and the Faddeev-Jackiw method which describes constr ained systems and produces nonstandard symplectic structures. We show that our model describes the superposition of a free motion in noncomm utative D = 2 spaces as well as the ''internal'' oscillator modes. We add a suitably chosen class of velocity-dependent two-particle interac tions, which is described by local potentials in D = 2 noncommutative space. We treat, in detail, the particular case of a harmonic oscillat or and describe its quantization. It appears that the indefinite metri c due to the third order time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposit ion of a subsidiary condition. (C) 1997 Academic Press.