Why is d'Alembert's principle not as applicable in physics as the principle of stationary action?

Question

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?

Answer 1

On one hand,

• The d'Alembert's principle is formulated in the framework of Newton's laws to start with, which limits its usefulness in other areas of physics.

• The d'Alembert's principle is more or less equivalent to Lagrange equations. The latter is arguably easy to apply.

On the other hand,

• the stationary action principle is versatile/applies to many different areas of physics.

• the stationary action principle does no cope well with e.g. dissipation and/or semi-holonomic constraints. However, this is often not an obstacle in fundamental physics. See also e.g. this Phys.SE post.