Optimal shape of a vertical rotating column

We determine the shape of the lightest rotating column that is stable against buckling, positioned in a constant gravity field, oriented along the column axis. In deriving the optimality conditions, the Pontryagin's principle was used. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. For this problem a variational principle and a first integral are formulated. Also a priori estimates of the cross-sectional area at the lower end are presented. The procedure is illustrated by three concrete examples. The problem treated here may be considered as a step in the dynamic optimization procedure of a heavy rotating column.