1. The problem statement, all variables and given/known data I figured out the first part of the question, proving why |t| equals 1, but I have trouble solving the next part of the problem. I expressed F(r(s)) in terms of theta, but I cannot solve for a, b, and c using the equation I derived. 2. Relevant equations Free energy minimization. Change of variable for a 2D geometry 3. The attempt at a solution I first attemtped to convert the given equation F(r(s))= ∫(ds 1/2*k(d2r/ds2)^2) using polar coordinate. In order to replace (d2r/ds2)^2, I differentiated dr/ds=(cosθ(s),sinθ(s)) respect to s. (d2r/ds2)^2=(-dθ/ds *sinθ(s), dθ/ds*cosθ(s))^2 = (dθ/ds)^2*sin^2(θ(s))+(dθ/ds)^2*cos^2(θ(s))=(dθ/ds)^2. Then to replace ds, ds=dθ*ds/dθ. Eventually I turned F(r(s))= ∫(ds 1/2*k(d2r/ds2)^2,s=0 to L) into F=∫(dθ 1/2*k(dθ/ds), θ=θ(0) to θ(l)) Is this correct?? Well, I thought it was a very simple and beautiful answer, but I could not solve the next problem using this equation. I do not know how I can minimize F=∫(dθ 1/2*k(dθ/ds), θ=θ(0) to θ(l)) when θ is given as the polynomial equation. By plugging in the polynomial equation, θ=b*s+c*s^2, a=0 due to initial condition θ(0)=0 dθ/ds==s+2cs F=∫(dθ 1/2*k(dθ/ds)=F=∫(dθ 1/2*k*(b+2cs)). Then I expressed 'b' in terms of c using the initial condition θ(l)=0; This is where I am stuck... Could you please help me .. I have been struggling with it all day long.