  We study the quantization of many-body systems in two dimensions in rotating
coordinate frames using a gauge invariant formulation of the dynamics. We
consider reference frames defined by linear and quadratic gauge conditions. In
both cases we discuss their Gribov ambiguities and commutator algebra. We
construct the momentum operators, inner-product and Hamiltonian in both types
of gauges, for systems with and without translation invariance. The analogy
with the quantization of QED in non-covariant gauges is emphasized. Our results
are applied to quasi-rigid systems in the Eckart frame.
