smallphi, thanks for your post. I now have a better understanding of this.

In fact Goldstein does say that if the constrains are holonomic, then it is possible to find sets of independent coordinates [itex]q_{j}[/itex] that contain constraint conditions implicitly in the transformation equations (that relate dependent coordinates to the generalized coordinates), so that [itex]q_{j}[/itex] is independent of [itex]q_{k}[/itex] for [itex]j \neq k[/itex]. So through your example, I now understand what this means.

Thanks again