generalized coordinate

From Physics wiki

Normally the configuration of a system can be described by specifying the positions $\mathbf{r}_i\,$ of all its components on some manifold, which is usually taken to be Cartesian coordinates on the Euclidean space $\mathbb{R}^3\,$ . These coordinates are democratic in that every component is treated on the same footing. On the other hand, the system may have a number of symmetries which may not be evident in the chosen coordinate system, or there may exist a number of constraints which introduce dependence among these coordinates. In such cases, we may parameterize the $m\,$ coordinates $\mathbf{r}_i\,$ in terms of $n\,$ variables $q_i\,$ which are termed generalized coordinates:

$\begin{matrix} \mathbf{r}_1=\mathbf{r}_1(q_1, q_2, ..., q_n, t),\\ \mathbf{r}_2=\mathbf{r}_2(q_1, q_2, ..., q_n, t),\\ \vdots\\ \mathbf{r}_m=\mathbf{r}_m(q_1, q_2, ..., q_n, t), \end{matrix}\,$

where we have allowed the coordinate transformation to be time dependent. If the system has $k\,$ degrees of freedom, there exist $3m - k\,$ constraint equations of the form $f_j(\mathbf{r}_i) = 0\,$ . The technique is most useful when these $k\,$ degrees of freedom are precisely described by $n\,$ generalized coordinates; that is, when $n = k\,$ . One then says that the constraints $f_j(\mathbf{r}_i) = 0\,$ are implicit in the coordinates $q_i\,$ .