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I want to set up the Euler-Lagrange-equations for the hanging rope-problem by using the Lagrange-formalism and Lagrange multipliers.

The rope is of length $l^*$ and is attached to the points $(-l,h),(l,h) $ with $2l<l^* $ and $0<h$.

The Potential energy is given by $$ U[y] = \rho g \int_{-l}^l y(x) \sqrt{1 + y'(x)^2} dx = \int_{-l}^l y(x) \sqrt{1 + y'(x)^2} dx $$ with $\rho g =1$. For the kinetic energy it holds that $T = 0$, yielding that the Lagrangian is simply $L_0=T-U=-U$.

So we want to find a smooth function that minimizes the functional $U$ under the constraint, that the graph of the function has length $l^*$. We can express this with

$$l^* = \int_{-l}^l \sqrt{1 + y'(x)^2} dx =:g[y]$$

as the length of the curve $y$ between $-l,l$. With this condition we will alter the objective-functional into $$ U[y] - \lambda g[y] = \int_{-l}^l (y(x)-\lambda)\sqrt{1+y'(x)^2}dx $$ Where we see the term in the integral as the constrained Lagrangian $L$, which we can plug into the Euler Lagrange-equation. But with the next step I don't know what I should do and why. I am also unsure, if it matters, if I encode the information about the length in the constraint, such as $g_l := g[y] - l^* = 0$.

Thanks!

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