Base-controlled mechanical systems and geometric phases

In this paper, we carry a detailed study of mechanical systems with configuration space $Q⟶Q\mathrm{/}G$ for which the base $Q\mathrm{/}G$ variables are being controlled. The overall system’s motion is considered to be induced from the base one due to the presence of general non-holonomic constraints. It is shown that the solution can be factorized into dynamical and geometrical parts. Moreover, under favorable kinematical circumstances, the dynamical part admits a further factorization since it can be reconstructed from an intermediate (body) momentum solution, yielding a reconstruction phase formula. Finally, we apply this results to the study of concrete mechanical systems.