Why use virtual displacement to make constraint forces vanish?

Question

Why do we use virtual displacement to vanish work done by constraint forces instead of the actual displacement?

Answer 1

1. In a nutshell, d'Alembert's principle states that a certain vector should be perpendicular to a constraint surface, i.e. perpendicular to all its tangent vectors, i.e. perpendicular to all infinitesimal virtual displacements. See also e.g. this related Phys.SE post.

2. Note that at this stage, we are just trying to find the EOMs of the physical system under investigation. At a later stage we will then try to solve the EOMs. We cannot use the actual displacement because we don't know the solution yet.

3. Btw, it should probably be stressed that the actual displacement never coincides with a virtual displacement, because the latter is frozen in time, cf. e.g. this & this related Phys.SE posts.

Answer 2

H. Goldstein, Classical mechanics, Chapter $1$ says

"Note that if a particle is constrained to a surface that is itself moving in time, the force of constraint is instantaneously perpendicular to the surface and the work during a virtual displacement is still zero even though the work during an actual displacement in the time $dt$ does not necessarily vanish."

I think that this is just an additive advantage of taking virtual displacement instead of actual displacement and that the main reason leads to the linear independence of the equation $$\sum_j \left\{\left[\frac{d}{dt}\left(\frac{\partial T}{\partial \dot q_j}\right)-\frac{\partial T}{\partial q_j}\right]-Q_j\right\}\delta q_j=0\tag{1.52}$$ which leads to the Lagrange's equations.

I used actual displacement $$dr_i=\sum_j \frac{\partial r_i}{\partial q_j}dq_j+\frac{\partial r_i}{\partial t}dt$$ instead of virtual displacement $$\delta r_j=\sum_j \frac{\partial r_i}{\partial q_j}\delta q_j\tag{1.47}$$ in the whole derivation of the Lagrange's equations and got an extra term depending on $\displaystyle\frac{\partial r_i}{\partial t}$ on the LHS of the equation $(1.52)$ which did not let me apply the logic of linear independence. That was two years ago. I can't find that notebook and I am too lazy to do it again. You try it yourself. Comment below, should you face any issues.

PS: Again, this is not from a verified source.