Suppose I have some points in a plane, and I want to integrate a form over a loop passing through all those points, how would I choose out of such loops the one which extremizes the integral?
Practically speaking, I'm trying to think if I have an engine which can operate in a certain range of pressure and volume, how I would let its state variable develop such that I get the maximum work. So, I guess answer would be a some sort of generalisation of the carnot cycle.
If the form is closed over the region then this simply doesn't matter. I'm looking for non trivial cases.
There may not be a solution to this. Given your $1$-form $\omega$, set $d\omega = f(x,y)\,dx\wedge dy$. If $f$ happens to be zero on your set of points and one of the sets $\{(x,y): f(x,y)>0\}$, $\{(x,y): f(x,y)<0\}$ is bounded then you can take the component of the boundary containing your set of points. Orient it appropriately to get the largest/smallest integral. However, if $f$ is nonzero at any of the points, you can easily see that you can perturb any closed curve passing through the points to make the integral larger/smaller.