When we learn the about the d'Alembert's principle we are introduced to the idea of virtual displacement. It also arises in Hamilton's principle and in Lagrangian mechanics.
But, what it really is? In Goldestein's Classical Mechanics book it states that is an infinitesimal change in position without violating system constraints and without the letting the time change (because constraints can be modified in this small amount of time). Basically: $$ \mathbf r_i(q_1, q_2, \dots q_k) $$ Then: $$ \delta \mathbf r_i = \sum_k \frac{\partial \mathbf r}{\partial q_k} \delta q_k $$ That's intuitive but a really vague explanation (the book does not bother to give a rigorous definition).
Then, we are introduced to virtual work: $$ \delta W = \sum_i \mathbf F_i\cdot \delta \mathbf r_i = 0 $$ That follows naturally from Newton's laws. However what does "virtual work" even mean? Isn't work defined via an integral?
When looking on wikipedia it now gives a definition via integrals and with virtual displacements, however I do not understand when in the demonstration says:
Suppose the force F(r(t) + εh(t)) is the same as F(r(t)).
Why would the force be the same in two different paths?
It would be clarifying to reconcile this two points of view (integral and discrete sum). And have an rigorous definition of what a virtual displacement is.
