I can start explaining the problem but a more quicker way would be to open this link: http://onlinelibrary.wiley.com/doi/10.1002/9783527627486.app2/pdf and check the paragraph resulting in expression (B.5). Note that I don't really care about the kinetic energy they talk about in this link, just the expression B.5 and the paragraph that precedes it. So it seems that as long as the constraints between the particles do not depend on time explicitly (scleronomic constraints), cartesian coordinates only depend on the generalized coordinates explicitly. However, then I have a question about what choices are allowed for the generalized coordinates. What if I pick another coordinate frame moving with a constant velocity relative to mine. And then I call those coordinates the generalized coordinates. Then obviously, no constraints are present but still the transformation between my cartesian coordinates and the generalized coordinates are explicitly time dependent. However in class we clearly stated that if no time dependent constraints are presents all the position vectors in our frame depend only on the generalized coordinates explicitly. Where am I thinking wrong?