Mechanical variation involving auxiliary functions

[YogiBear]

Homework Statement

A chain of length L and uniform mass per unit length ρ is suspended in a uniform gravitational field. The potential energy U[y] and length l[y] functionals of the chain can be written in terms of y(x) as follows:

U[y] = ρg*Int(y(1+y'^2)^1/2 dx) l[y] = Int((1+y'^2)^1/2) dx

where (x = 0, y = 0) and (x = a, y = 0) are the end points of the chain. Deduce the shape of the chain using the following procedure:
Everything after this point are guidelines.
- Write down an appropriate auxiliary functional for the problem (i.e. a functional that includes a Lagrange multiplier). - Find the first integral E(y, y′ ) associated with the auxiliary functional. - Use the first integral to find y ′ in terms of y, the Lagrange multiplier and the conserved quantity corresponding to the first integral. - Find the general solution to the first-order ordinary differential equation for y(x) found in the previous step. Your answer should contain two constants of integration (one of them will be the conserved quantity). - Use the boundary conditions to eliminate the second constant of integration and the Lagrange multiplier. - Calculate l[y] and find the length L of the chain in terms of the physical parameters ρ, g, a and the conserved quantity

Homework Equations

Sλ[q(t)] = S[q(t)] − λF[q(t)]

The Attempt at a Solution

Well i tried to create an auxiliary equation in which my alpha(y,y') = pgy*(1+y'^2)^1/2 - λ(1+y'^2)^1/2

Which i am 100% sure to be wrong, as when i follow through I get to the point where I get everything canceled out and am left with 1 = 0.
If someone were to provide me with a corrected auxiliary function i believe i should be able to follow the instructions. But if you would like to talk me through the rest of the question i would appreciate it.

 

[mark726]

Thank you for your post. I am a scientist who has experience with solving problems involving auxiliary functionals, so I would be happy to help you with this question.

First of all, your attempt at creating an auxiliary functional is not entirely wrong, but it can be improved. Let me explain why. The purpose of an auxiliary functional is to introduce an additional constraint into the problem, which will help us find a solution that satisfies both the original functional and the constraint. In this case, we know that the length of the chain is fixed at L, so we can write an auxiliary functional that includes the constraint L[y] = L. This can be done by using a Lagrange multiplier, which I will denote by λ. So, our auxiliary functional should look like this:

Sλ[y(x)] = S[y(x)] - λ(L[y] - L)

Now, let's move on to finding the first integral E(y,y') associated with this auxiliary functional. This step involves taking the variation of the auxiliary functional with respect to y(x) and setting it equal to 0. This will give us an equation involving y and y', which we can then solve for y' in terms of y, λ, and the conserved quantity corresponding to the first integral. In this case, the first integral can be written as follows:

E(y,y') = pgy*(1+y'^2)^1/2 - λ(1+y'^2)^1/2 + μ

where μ is the conserved quantity. Note that I have included a constant of integration μ, which will be useful in finding the general solution for y(x).

Next, we need to solve the first-order ordinary differential equation for y(x) that we obtained in the previous step. This can be done by separating variables and integrating on both sides. The result will be a general solution for y(x) that contains two constants of integration (one of them being μ).

Now, we can use the boundary conditions (y(0) = 0 and y(a) = 0) to eliminate the second constant of integration and the Lagrange multiplier. This will give us an expression for μ in terms of the physical parameters ρ, g, and a. We can then use this expression to find the length L of the chain, which will be in terms of the physical parameters and the conserved quantity μ.

I hope this helps you in solving the problem. If you have any