So, I've been reading Thornton and Marion's "Classical Dynamics of Particles and Systems" and have gotten to the chapter on the calculus of variations. In trying the end of chapter problems, I find I'm totally baffled by 6-9: given the volume of a cylinder, find the ratio of the height to the radius that minimizes the surface area. Now, just using elementary calculus, the solution is easy enough: take the volume, solve for the height, plug that back in the equation for area so you only have the radius, differentiate and set the result to zero, and solve. Voila, the ratio is 2. That's not my problem... how would you set this up to solve it using the Euler-Lagrange equation? I''m not really seeing how it fits... although it seems to me that it must. I've tried using the ratio as the independent variable, and the radius... but I somehow am not imposing the condition that the volume must be specified. (When I tried, I ended up with a cylinder of zero surface area... which I suppose *is* minimal, but...) Thanks.