Elastic Stability

Abstract

In the lecture we have seen that conjugate points are defined by δ²F = 0. The interval between them is the smallest one for which δ²F vanishes. δ²F vanishes in any case if at least two neighbouring solutions intersect each other at the endpoints of the considered interval. But it is unknown for most of the problems whether such neighbouring solutions must always exist at conjugate points. This is, in the lecture, proved only for LAGRANGE-variations of one-dimensional problems without constraints, where the functional F depends on z and z′ only. So we are sure that we can find out the conjugate points by looking for neighbouring solutions only in those cases where these conditions are satisfied.