I have a functional that I would like to extremise with the constraint that the solution has to be antisymmetric. I am unsure if my understanding of Lagrange multipliers is correct. The statement of the problem is
Minimize $\,F[f(x,y)]\,$ w.r.t $\,f(x,y)\,$ s.t $\,f(x,y) = -f(y,x)$.
The form of $\,F[f(x,y)]\,$ is known and I can minimise it without the constraint just fine. My problem is with enforcing the constraint.
My first attempt:
Let $\mu(x,y)$ be a $\textit{continuous}$ Lagrange multiplier and
$\mathcal{L}[f(x,y)] = F[f(x,y)] + \int [f(x,y)+f(y,x)]\mu(x,y)dxdy$
Then set the functional derivatives equal to zero
$\begin{align}\frac{\delta \mathcal{L}}{\delta\mu(x,y)} &= f(x,y) + f(y,x) =0 \\ \frac{\delta \mathcal{L}}{\delta f(x,y)} &= \frac{\delta F}{\delta f(x,y)} + \mu(x,y) + \mu(y,x) = 0\end{align}$
where the final term $\mu(x,y) + \mu(y,x)$ comes from the Euler-Lagrange equations applied to the constraint term. Is this the correct way to apply lagrange multipliers in this context?
Further problem:
For my application, it is convenient for my Lagrangian to be of the form
$\mathcal{L}[f(x,y)] = F[f(x,y)] + \int f(x,y)[f(x,y)+f(y,x)]\mu(x,y)dxdy$
Note the extra $f(x,y)$ in the constraint. I then have
$0=\dfrac{\delta\mathcal{L}}{\delta f(x,y)} = \dfrac{\delta F}{\delta f(x,y)} +2f(x,y)\mu(x,y) + f(x,y)(\mu(x,y)+\mu(y,x))$
Is it ok to modify the constraint in this way?
Thanks