Solve Catenary Problem: Minimize Potential Energy

[BubblesAreUs]

Homework Statement

Suppose we have a rope of length L and total mass M. Suppose we x its ends at points
(x_A; y_A) and (x_B; y_B). We want to determine the shape the rope makes, hanging under the
influence of gravity. The rope is motionless, with a shape parametrised by y(x) or equivalently,
x(y), where x denotes the horizontal coordinate and y the vertical one. We are looking for the
shape which minimises the potential energy of the rope.Image below
[/B]

Homework Equations

I'm guessing

ds = sqrt ( dx^2 + dy^2) can be used.

The Attempt at a Solution

[/B]
Integrate ds over s, and thus it is...

integral ds = S [ from Yb to Ya]

Xb and Xa would be zero as the horizontal length does not change.

As you can see...I'm a bit confused. I don't know how to parametise dx and dy, or can I just use a polar coordinate system?

 

[Orodruin]

Integrate ds over s, and thus it is...

This just gives you the length of the rope, which you know and should impose as a constraint. You need to find an integral that describes the potential energy and minimize it under that constraint.

 

[Dr.D]

The integral for the potential energy was given as eq(1) of the problem statement. All that is necessary is to minimize that. This is a well known problem, written up in countless places in the literature.

Your guess,

ds = sqrt ( dx^2 + dy^2)

was already incorporated in writing eq(1).