What is difference between variations of the work and virtual work?

Question

I really want to know whether or not both equations are the same mathematically. I think that they are the same, I just want to be sure.

(Reference: this website.)

Answer 1

The bottom line:

1. Virtual work is work of a virtual displacement, cf. e.g. this and this Phys.SE posts. In particular, a virtual displacement is frozen in time.

2. In contrast, actual work and an actual displacement in physics happen as time evolves forward.

Answer 2

The first one is the definition of the virtual work (or variation of the work, or virtual variation of the work) and $F$ denotes generalized forces, which includes also torque. The second one is an application of the virtual work principle for the specific case of static equilibrium.