Generalized Coordinates

Let the $q_i$, for $i=1,{\cal F}$, be a set of coordinates which uniquely specifies the instantaneous configuration of some dynamical system. Here, it is assumed that each of the $q_i$ can vary independently. The $q_i$ might be Cartesian coordinates, or polar coordinates, or angles, or some mixture of all three types of coordinate, and are, therefore, termed generalized coordinates. A dynamical system whose instantaneous configuration is fully specified by ${\cal F}$ independent generalized coordinates is said to have ${\cal F}$ degrees of freedom. For instance, the instantaneous position of a particle moving freely in three dimensions is completely specified by its three Cartesian coordinates, $x$, $y$, and $z$. Moreover, these coordinates are clearly independent of one another. Hence, a dynamical system consisting of a single particle moving freely in three dimensions has three degrees of freedom. If there are two freely moving particles then the system has six degrees of freedom, and so on.

Suppose that we have a dynamical system consisting of $N$ particles moving freely in three dimensions. This is an ${\cal F}= 3\,N$ degree of freedom system whose instantaneous configuration can be specified by ${\cal F}$ Cartesian coordinates. Let us denote these coordinates the $x_j$, for $j=1,{\cal F}$. Thus, $x_1, x_2, x_3$ are the Cartesian coordinates of the first particle, $x_4, x_5, x_6$ the Cartesian coordinates of the second particle, etc. Suppose that the instantaneous configuration of the system can also be specified by ${\cal F}$ generalized coordinates, which we shall denote the $q_i$, for $i=1,{\cal F}$. Thus, the $q_i$ might be the spherical coordinates of the particles. In general, we expect the $x_j$ to be functions of the $q_i$. In other words,

$$x_j = x_j(q_1,q_2,\cdots,q_{\cal F}, t)$$ (592)

for $j=1,{\cal F}$. Here, for the sake of generality, we have included the possibility that the functional relationship between the $x_j$ and the $q_i$ might depend on the time, $t$, explicitly. This would be the case if the dynamical system were subject to time varying constraints. For instance, a system consisting of a particle constrained to move on a surface which is itself moving. Finally, by the chain rule, the variation of the $x_j$ due to a variation of the $q_i$ (at constant $t$) is given by

$$\delta x_j = \sum_{i=1,{\cal F}} \frac{\partial x_j}{\partial q_i}\,\delta q_i,$$ (593)

for $j=1,{\cal F}$.