Lack of Constraint equations I was trying to find how a uniform string of length $L$ fixed  at a point (I assumed $(0,0)$) bends under gravity. I tried to minimise the potential energy within the constraint of the length of the string. The action I got was 
$$ S = \int^{x_f}_{0}{ \bigg(\mu g y \sqrt{1+ \bigg(\frac{dy}{dx}\bigg)^2} - \lambda \sqrt{1+ \bigg(\frac{dy}{dx}\bigg)^2}\bigg) dx}$$
where $\lambda$ is a Lagrange multiplier and $\mu$ is the linear mass density. The upper limit $x_f$ is not really known. I solved the Beltrami equation to find $y(x)$. I got
$$ y(x)= \frac{1}{\mu g} \bigg(c_1 \cosh{\bigg(\frac{c_1x+c_1 c_2}{\mu g}\bigg)}- \lambda \bigg)$$
Where $c_1$ and $c_2$ are arbitrary constants. There must be three constraint relations to find $c_1 , c_2$ and $\lambda$. One of them is that $(0,0)$ is fixed. The other relation should be that the length is always constant. But to apply this one needs to know the coordinate of the tip of the string. 
Since we do not know the coordinate of the tip, what other two constraint relations should this rod satisfy? 
If there aren't any constraint relations, have I done something wrong or am I missing something very obvious?
 A: If there aren't any constraint relations, have I done something wrong or am I missing something very obvious?
Yes I believe you have done something wrong, and something is obviously incorrect if I understand what your $y(x)$ function means.
I don’t know anything about the Lagrangian but if $y(x)$ is the downward deflection of the rod as a function of $x$, the distance from the fixed end of the rod, then the equation can’t be correct.
Refer to the diagram below showing the deflection of a uniformly loaded cantilever beam. For your case, the uniform load $w$ is the weight per unit length of the beam, or $\frac {mg}{L}$.
The second equation shows the deflection as a function of the distance from the fixed end of the beam (rod, etc.). If this analogous to your $y(x)$ and your $μg$ is analogous to my $mg/L$, then your equation says the deflection is inversely related to the load (weight) of the beam. This would be obviously incorrect.
In addition, there are two variables $E$, the modulus of elasticity (a property of the material,) and $I$, the moment of inertia (a function of the cross section area) that are essential in determining the deflection. So they would be constraints. I don’t see either in your equation, unless they are somehow related to your $C_1$ and $C_2$ or are buried $λ$. 
Hope this helps.

