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 P: 1,780 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 $q_{j}$ that contain constraint conditions implicitly in the transformation equations (that relate dependent coordinates to the generalized coordinates), so that $q_{j}$ is independent of $q_{k}$ for $j \neq k$. So through your example, I now understand what this means. Thanks again