1. d'Alembert's principle $\to$ Lagrangian mechanics
Let's consider a dynamical system with N particles & k constraints. $\delta x^i \: \left( i=1,2,\cdots,3N \right) $ is a virtual displacement, and let $q^i \left( i=1,2,\cdots,3N-k \right) $ be a set of generalized coordinates. Virtual displacement can be represented by
$$\delta x^i = \frac{\partial x^i}{\partial q^j} \delta q^j$$
There is for sure no time-dependence since virtual displacement is just an instant phenomena. d'Alembert's principle indicates that virtual work done by constraint force $R_i$ in conjugation with $\delta x^i$ is always zero :
$$ R_i \delta x^i = 0$$
Alternatively, we can express the same principle using external force $F_i^{(a)}$ like below :
$$ (F_i^{(a)}-\dot{p}_i) \delta x^i =0$$