Solve the hanging rope shape using the variational principle? I know how to write down the local ordinary differential equation (ODE) via Newton's force law, balancing between the left and right rope tension and the gravity exerted on a infinitesimal piece of rope.
The solution is Catenary which has the form
$$
y (x)= a \cosh \left(\frac{x}{a} \right) = \frac{a}{2}\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right)
$$
with appropriate boundary conditions. But I only aware the way to solve from ODE.
Could we solve the hanging rope shape using the variational principle? This means that we input the shape of the rope minimize the potential energy $V$ (because there is no kinetic $T$) for lagrangian $L= T-V$ (question 1: but this seems maximize the $L$ strangely?).
Suppose the length of the role is fixed say $\ell$. The two hanging points out horizontal apart at $x=-l$ and $x=l$.
Then we have a constraint on the length of the role is fixed say $\ell$. Then we have a constraint
$$
\ell = \int_{-l}^l \sqrt{1 + y'(x)^2} dx  
$$
Then we minimize potential energy $V$ (but this maximizes the lagrangian $L$ strangely?)
$$
V= \rho g \int_{-l}^l y(x) \sqrt{1 + y'(x)^2} dx 
 $$
with $ \rho$ the constant density and gravity constant $g$.
I suppose we are solving the eq of motion from Variational Principle with a Constraint like
$$\boxed{
V - \lambda(\ell - \int_{-l}^l \sqrt{1 + y'(x)^2} dx )= \rho g \int_{-l}^l y(x) \sqrt{1 + y'(x)^2} dx 
 - \lambda(\ell - \int_{-l}^l \sqrt{1 + y'(x)^2} dx )}
$$
How do we obtain $y(x)$ then by minimization or maximization ?
See these two related questions show no answers (!!!):
https://math.stackexchange.com/q/3490286/
https://math.stackexchange.com/q/3498723/
 A: The catenary is a problem in statics, so the variational apparatus of dynamics doesn't apply.
The variational form for the hanging chain is about finding the curve that is a minimum for the potential energy. (The middle of the chain tends to pull the sides inward. But when the sides are pulled inward they are raised.)
So: for the catenary the variational problem is to evaluate the potential energy:
$\rho$ = mass per unit of length
$g$ = gravitational acceleration
$$ U = \rho g \int y \sqrt{1 + (\tfrac{dy}{dx})^2} \ dx  $$
(In dynamics what is inserted in the Euler-Lagrange equation is Hamilton's action. In the case of the catenary what is inserted in the Euler-Lagrange equation is the potential energy $U$.)
There is a treatment of the catenary in terms of minimizing potential energy in a blog post on a blog titled 'Conversation of momentum (no that is not a typo)'
Also recommended: discussion by Preetum Nakkiran who uses catenary minimizing potential energy as motivating case for a derivation of the Euler-Lagrange equation that uses only local reasoning (instead of using integration by parts)
