Suppose we are given a 2D force vector field $F(x,y)$ (position dependent only, no time dependency) and we're given a path C in the plane, and we compute $\int_C F \cdot ds$ and get zero. I was told (without explanation) that this means the force cannot move any object from the start of the path to the end of that path. Question 1: Is this (generically) true? (That is, generally speaking, one can think of it this way, and why, or is it simply false.) Question 2: What if we compute $\int_C F \cdot ds > 0$. Would this mean it "could" push an object from the beginning to the end of the path?
Someone could come up with a better example, but here is an explicit question one could ask: Supposing $C$ was the parametrization of some track, if $\int_C F \cdot ds = 0$, and we put a cart (of any mass) at the beginning of the track, that force $F$ could not move the cart from the start of the track to the end of the track? If $\int_C F \cdot ds > 0$, then there would be a cart we could place at the beginning of the track and the force would move it to the end?