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

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?