We study mechanical systems where part of the degrees of freedom are being controlled in a known way and determine the motion of the rest of the variables due to the presence of constraints/conservation laws. More concretely, we consider the configuration space to be a G-bundle Q \to Q/G in 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 or conservation laws. We show that the overall solution can be factorized into dynamical and geometrical contributions (geometric phase), yielding a so called reconstruction phase formula. Finally, we apply this results to the study of concrete mechanical systems like a self-deforming satellite in space.