Let a cable of length $L$ and uniform mass density $\rho=1$. The cable is hanging between two columns of height $h>>1$ separated by a distance $d\in(0,L)$. Let a point mass $m>0$ be attached to the cable at an arc distance $\ell\in(0,L)$ from the left edge. You may assume $g=1$ for the free acceleration. Find the extremals of the functional which minimizes the potential energy under the corresponding constraint. Show that there is only one extremal in the case of $\ell=\frac{L}{2}$ when the cable is assumed to be symmetric w.r.t. its center.
For this problem, I found it convenient to work with an arclength formulation, i.e. I use an arclength parameterization for the cable $(x(s),y(s))$ s.t. $\dot{x}^2+\dot{y}^2=1\ \forall s\in[0,L]$. Thus I obtained the functional $$J[y]=\int_0 ^Ly(s)ds+my(\ell)$$ subject to the constraint $$\int_0 ^L\sqrt{1-\dot{y}^2}ds=d,\ y(0)=y(L)=h.$$ Where the integral constraint was obtained by requiring $x(L)-x(0)=d$. But it is still unclear to me what is the correct adjustment of the Euler-Lagrange necessary condition or the du Bois-Reymond necessary condition to such a functional, even after trying to adjust the Euler-Lagrange derivation to this case.
Any help would be appreciated.