Any textbook in classical mechanics will tell you that there are two different routes one can follow to derive the Euler-Lagrange equations:
- Route 1: Write d'Alembert's principle in the form $\sum_{i=1}^N(\mathbf{F}^{(A)}_i-\frac{\mathrm{d}\mathbf{p}_i}{\mathrm{d}t})\cdot\delta\mathbf{r}_i=0$, where $N$ is the number of particles in the system and the summation is performed over all particles. Re-writing this in terms of generalized coordinates and considering generalized forces of the form $Q_j=-\frac{\partial U}{\partial q_j}$, one arrives at the Euler-Lagrange equation.
- Route 2: Consider the functional $S(\mathbf{r}(t),\dot{\mathbf{r}}(t),t)=\int_{t_2}^{t_1}L(\mathbf{r}(t),\dot{\mathbf{r}}(t),t) \mathrm{d}t$, called action. The Euler-Lagrange equations are then derived by requiring $\delta S=0$, that is, the stationarity of the action.
While route 2, called the principle of stationary action, is valid in all of physics, from Newtonian mechanics to Quantum Field Theory, route 1 is not used outside of classical point particle mechanics. Why this is so? Why hasn't d'Alembert's principle been generalized to apply outside of classical point particle mechanics?
