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?


1 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,


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