Generalized force

Remarks

Equivalently, $\sum_i Q_i\, \delta q_i$ is the virtual work $\sum_k F_k \cdot \delta x_k$ of the applied forces for the virtual displacement $\delta x_k = \sum_i \frac{\partial x_k}{\partial q_i}\, \delta q_i$ at fixed time. Example: a pendulum of length $\ell > 0$ and mass $m$ has $x(\theta) = (\ell\sin\theta, -\ell\cos\theta)$ and force $F = (0, -mg)$, so $Q_\theta = F \cdot (\ell\cos\theta, \ell\sin\theta) = -mg\ell\sin\theta$. If the forces derive from a potential, $F_k = -\nabla_{x_k} V$, then by the chain rule $Q_i = -\partial V / \partial q_i$. Non-example: the sum $\sum_k F_k \cdot x_k$ is not a generalized force, because it does not weigh each force by the displacement $\partial x_k / \partial q_i$. For the pendulum it gives $mg\ell\cos\theta$ instead of $-mg\ell\sin\theta$. Forces of ideal constraints do not contribute to the generalized forces, because they are orthogonal to all virtual displacements.