To derive Lagrange's Equations we need the principle of virtual work first. This principle states that whenever a system of $K$ particles is constrained to a submanifold $\mathcal{M}\subset \mathbb{R}^{3K}$ and if the forces of constraint are perpendicular to $\mathcal{M}$ then in equilibrium holds
$$\langle \mathbf{F}(c(t)), \mathbf{v}\rangle = 0 \qquad \forall \ \mathbf{v} \in \mathcal{M}_{c(t)}$$
Where $c$ is the path of evolution of the system. Now, for this to hold we need that part "the forces of constraint are perpendicular to $\mathcal{M}$" or in other words, they are workless in virtual infinitesimal displacements. Now, why is this a reasonable approach?
In the books I've been looking at the justification is always "because it holds for rigid bodies so we mimic this to other cases" but this seems not natural. Is there a deeper intuition in restricting the problems being studied to those satisfying this condition?
