On a cubic variational problem

Abstract

An extremal curve of the simplest variational problem is a continuously differentiable function. Hilbert’s differentiability theorem provides a condition that guarantees the existence of the second derivative of an extremal curve. It is desirable to have a simple example in which the condition of Hilbert’s theorem fails to hold true and an extremal curve is not twice differentiable. In this paper, we analyse a cubic variational problem with the following properties. The functional of the problem is neither bounded from above nor bounded from below. There exists an extremal curve of this problem that is obtained by pasting together two different extremal curves, and that is not twice differentiable at the sewing point. Despite this unfavourable situation, an attempt to apply the method of steepest descent (in the form proposed by V. F. Demyanov) to this problem is made. It appears that the method converges to the extremal curve provided one chooses a suitable step size rule. Refs 2. Figs 6. Table 1.