I am trying to use caluclus of variations to extremise the shape of a contour (closed path) in the $xy$-plane that defines an area on this planar surface, within a given region. I am having difficulty applying/adapting the typical theory, which I have learned in a context that doesn't involve contour integrals and vector fields, to this problem. I hope to get some insight on how I should proceed for this case.
Problem description: In the $xy$-plane, say in the bounded region $x\in[x_{a},x_{b}], y\in[y_{a},y_{b}]$, there exists a closed contour (loop) of some shape $g(x,y)=0$ that we are free to tweak. The planar surface area enclosed by this contour is $\boldsymbol{A}=A\boldsymbol{\hat{z}}$. Now, a scalar field $W=W(x,y)$, with no variations in $z$ (i.e. $\partial/\partial z=0)$ passes through this plane and our task is to extermise the functional
$$ J(y,y_{x}):=\iint_{A} W(x,y) dx dy,$$
by finding the optimal shape for the contour $g(x,y)$ that defines the area $A$ (symbol $y_{x}$ denotes the derivative of $y$ in $x$). One would typically expect the solution to be parameteric in form, since the shape is a closed loop within the given region. Bold symbols denote vectors.
So, to proceed with this, I faced two problems and I tried to attack them as follows (hoping to put the problem into a more familiar variational form):
(1) Needed to convert the surface integral to contour (path) integral, so that I can find the relation between $x$ and $y$ on the path, to give $g$. So, I used Stokes' theorem, by defining some vector $\boldsymbol{F}$, such that $W\boldsymbol{\hat{z}}=\nabla \times \boldsymbol{F}$, and $J$ becomes
$$J:=\iint_{A} W(x,y) dx dy\equiv\iint_{A} W\boldsymbol{\hat{z}}\cdot \boldsymbol{\hat{z}} dA=\iint_{A} (\nabla \times \boldsymbol{F})\cdot \boldsymbol{\hat{z}} dA=\oint_{g}\boldsymbol{F}\cdot\boldsymbol{ds},$$
where $\oint_g$ denotes the integral around the contour [say we take $\int_{x}$ while putting $y$ according to $g(x,y)=0$ inside the integral; or we use some other parameter ($t$) for integration and continue parameterically as $x=x(t), y=y(t)$], and $\boldsymbol{ds}=ds\boldsymbol{\hat{s}}$ is the element of length along the path of the contour.
The problem faced here is: how to find $\boldsymbol{F}$ in general, in terms of the given $W$ field, to proceed?
(2) Then we can write $ds=dx\sqrt{1+(d y/d x)^{2}}=dx\sqrt{1+y_{x}^{2}}$.
But the problem here is: how do we find/write the unit vector $\boldsymbol{\hat{s}}$ along the path in terms of other variables? For example, do we say the $\boldsymbol{\hat{s}}$ is perpendicular to $\nabla g$, so that $\boldsymbol{\hat{s}}\cdot \nabla g=0$? But how can this be written down in convenient form to substitute for $\boldsymbol{\hat{s}}$ while $g$ is still general (unknown)?
How do we proceed so that we arrive at a standard variational form that allows us to then use the Euler-Lagrange equation and find $g$?