Does there exist any continuum or fluid mechanics / field theory analogue for d'Alembert's principle?

Question

Lanczos's book the "Variational Principles of Mechanics" states that

D'Alembert's principle gives a complete solution of problems of mechanics. All the different principles of mechanics are merely mathematically different formulations of d'Alembert's principle... This principle is more elementary than the later variational principles.

I also note that in Goldstein (and this SE answer) for example, it is used in obtaining Lagrangian mechanics from Newtonian mechanics.

However, it is not mentioned when taking the continuum limit to pass from discrete mechanics of point particles to go to continuum mechanics; whereas we can go from a Lagrangian to a Lagrangian density easily (i.e. we can take the continuum limit of the usual Lagrangian formulation but we do not go from the "key identity" of this SE answer).

Is there any obvious reason why? I was also vaguely imagining one might be able to follow some connection by swapping between the Lagrangian/Eulerian specifications of fluid mechanics, and that the "force on the $i$th particle" would turn into some kind of pressure or stress term or tensor.

Answer 1

I believe that finding an action priciple for the Eulerian description (as opposed to the Lagrangian "follow the particle" description) of fluid mechanics is very much a 20th-century discovery. For example, for irrotational barotropic flow in which ${\bf v}=\nabla\phi$, we can vary $\rho$ in the action functional $$ S[\rho, \phi] = \int \left\{ \rho\frac{\partial \phi}{\partial t} +\frac 12 \rho |\nabla \phi|^2 +u(\rho)\right\} d^3 x dt $$ to get, Bernoulli's equation
$$ \frac{\partial \phi}{\partial t}+ \frac 12 \rho |\nabla \phi|^2+h(\rho)=0,\quad h(\rho)= \frac{\partial u}{\partial \rho} $$ whose gradient gives us the Euler equation. By varying $\phi$ we get particle/mass conservation
$$ \frac{\partial \rho}{\partial t}+ \nabla \cdot(\rho{\bf v})=0. $$ That mass (or particle number) conservation follows from a variation shows that we cannot derive this from a standard many-particle Lagrangian as that formalism has particle number conservation built in.

I think that this formalism is due to Harry Bateman in the 1929's, but I am not sure. I do know that C C Lin in 1965 used the Clebsch decomposition $$ {\bf v}= \nabla \phi +\chi \nabla \psi $$ to write down actions for more general fluid flows. It is really a semiclassical version of of the quantum particle/field duality as $\phi$ is the phase of the field operator $\varphi$ that creates or destroys scalar particles.

So the bottom line is that it is not easy to go from Lagrange's many particle equations equation to Euler's continuuum fluid equations by simple replacement of sums by integrals.