Finding the shape of a hanging rope

[sap]

Homework Statement

given two poles at a distance between them and a rope that it's length is bigger than the distance between them, describe the shape of the rope.

Relevant Equations

no relevent equations

i started to think to maybe do an integral to find the minimum area, and then I thought that the area itself is not sufficient because there is more material depending on the slope. so I thought to do an integral depending on the length instead of x.
$dh^{2}=dx^{2}+dy^{2}$

$\int{}f(x)dh= \int{}f(x)\sqrt{dx^{2}+dy^{2}}$
$dy=\frac{dy}{dx}dx=dx f'(x)$
$\int{}f(x)\sqrt{dx^{2}+f'(x)^{2}dx^{2}}=\int{}f(x)\sqrt{f'(x)^{2}+1} dx^{2}$
how can I get a minimum to solvw this question?

 

[Kyuubi]

You can edit your post to fix latex issues rather than constantly making an entirely new post.

 

[docnet]

Look up the catenary and the principle of least action.

 

[Orodruin]

You only have an integral describing the potential energy but without restrictions on the string length. In order to find an actual minimum of the potential energy, you will furthermore need to impose additional constraints for the string length. There are several ways in which you can do this.

Look up the catenary and the principle of least action.

Nothing is moving here so the principle of stationary* action is not really needed. Only minimizing the potential energy.

* The ”principle of least action” is a misnomer. It is more accurate to use ”stationary” as the solution may be a maximum or saddle point as well.