Generalized forces

The work done on the dynamical system when its Cartesian coordinates change by $\delta x_j$ is simply

$$\delta W = \sum_{j=1,{\cal F}} f_j\,\delta x_j$$ (7.3)

Here, the $f_j$ are the Cartesian components of the forces acting on the various particles making up the system. Thus, $f_1, f_2, f_3$ are the components of the force acting on the first particle, $f_4, f_5, f_6$ the components of the force acting on the second particle, and so on. Using Equation (7.2), we can also write

$$\delta W = \sum_{j=1,{\cal F}} f_j\sum_{i=1,{\cal F}}\frac{\partial x_j}{\partial q_i}\,\delta q_i.$$ (7.4)

The preceding expression can be rearranged to give

$$\delta W = \sum_{i=1,{\cal F}} Q_i\,\delta q_i,$$ (7.5)

where

$$Q_i = \sum_{j=1,{\cal F}} f_j\,\frac{\partial x_j}{\partial q_i}.$$ (7.6)

Here, the $Q_i$ are termed generalized forces. A generalized force does not necessarily have the dimensions of force. However, the product $Q_i\,q_i$ must have the dimensions of work. Thus, if a particular $q_i$ is a Cartesian coordinate then the associated $Q_i$ is a force. Conversely, if a particular $q_i$ is an angle then the associated $Q_i$ is a torque.

Suppose that the dynamical system in question is conservative. It follows that

$$f_j = -\frac{\partial U}{\partial x_j},$$ (7.7)

for $j=1,{\cal F}$, where $U(x_1,x_2,\cdots,x_{\cal F},t)$ is the system's potential energy. Hence, according to Equation (7.6),

$$Q_i = - \sum_{j=1,{\cal F}} \frac{\partial U}{\partial x_j}\,\frac{\partial x_j}{\partial q_i} = - \frac{\partial U}{\partial q_i},$$ (7.8)

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