  In this paper, we carry a detailed study of mechanical systems with
configuration space $Q\longrightarrow Q/G$ for which the base $Q/G$ 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.
