Optimal shape of a strongest inverted column

By using Pontryagin's maximum principle we determine the shape of the strongest column positioned in a constant gravity field, simply supported at the lower end and clamped at upper end (with the possibility of axial sliding). It is shown that the cross-sectional area function is determined from the solution of a nonlinear boundary value problem. A variational principle for this boundary value problem is formulated and two first integrals are constructed. These integrals lead to an a priori estimate of the value of one the missing initial condition and to the reduction of the order of the system. The optimal shape of a column is determined by numerical integration.