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.)
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$\begingroup$ They are different because the second includes torques (moments), while the first does not. But they apply principle of virtual work. The second equation gives the conditions for static equilibrium, hence is describing a specific physical situation. The first equation is more of a definition. $\endgroup$– Peter DiehrCommented Jul 26, 2016 at 12:17
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2 Answers
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The bottom line:
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.
In contrast, actual work and an actual displacement in physics happen as time evolves forward.
answered Jun 5, 2019 at 12:21
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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.
