The tallest column problem: New first integrals and estimates

We analyze the problem of finding the shape of the tallest column. For the system of equations that determine the optimal shape we construct a variational principle and two new first integrals. From the first integrals we are able to determine, analytically, the size of the cross-sectional area of the optimal column at the bottom, as well as the corresponding bending moment and curvature of the elastic line. Our result for critical load is compared with the results obtained by other methods.