Calculus of variations for suspended rope

[aaaa202]

So perhaps you know this classical problem: A rope is suspended between two endpoints x=±a. Find what function describing the shape of the rope that will minimize its potential energy.
The example is worked through in my book but I have some questions:
The solution assumes uniform linear density which makes sense. However, in calculating the potential energy of the rope they write:
U(y) = -ρg∫y ds , ds = √(1+y'²)dx

First question:
I want to discuss this ds. What physically justifies using this as your "weight" for the average potential energy?

The book then proceeds to solve the problem using lagrange multipliers since the constraint is that the length of the rope remains fixed L. However, my book never mentions anything about the constraint y(-a)=y(a)=0. Why does this constraint not have to be taken into account?

 

[clamtrox]

However, in calculating the potential energy of the rope they write:
U(y) = -ρg∫y ds , ds = √(1+y'²)dx

First question:
I want to discuss this ds. What physically justifies using this as your "weight" for the average potential energy?

The infinitesimal mass element of the rope is dM = ρ ds, ds = √(dx²+dy²), right? Then you have dU = -dM g y like usual

The book then proceeds to solve the problem using lagrange multipliers since the constraint is that the length of the rope remains fixed L. However, my book never mentions anything about the constraint y(-a)=y(a)=0. Why does this constraint not have to be taken into account?

You take this into account when you solve the resulting differential equation. They give the boundary conditions.