# Homework Help: Mechanical variation involving auxiliary functions

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1. Jan 24, 2016

### YogiBear

1. The problem statement, all variables and given/known data
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

2. Relevant equations
Sλ[q(t)] = S[q(t)] − λF[q(t)]

3. 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 cancelled 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.

2. Jan 30, 2016