Chain shape (Euler-Lagrange equations)

[neworder1]

A chain with uniform linear density d and length L is tied at two ends to the ceiling. How to find its shape using Euler-Lagrange equations? (I know it can be done with other methods, but I want to know how to do it using E-L).

 

[siddharth]

First of all, you need to know what quantity is to be minimized. Next, you'll also have to consider the constraint (in the form of an integral) that the total length of the chain is L. So, use a lagrange undetermined multiplier so that you have the functional [itex] g = f + \lambda f_1[/tex], where f is the integrand which needs to be minimized and $f_1$ is the constraint. If you apply the Euler Lagrange equations to g, you'll be able to get the shape of the chain. To find $\lambda$, you'll need to use the constraint. Can you solve it from here?

 

[Dick]

Actually, I don't think it can be done without E-L. How would you do it?

 

[neworder1]

Yes, it can be done without E-L "manually", i.e. by writing forces, angles etc., but it's a very tedious way.