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.