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 sufficient condition for 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 is violated and an extremal curve is not twice differentiable.

In this paper, a cubic variational problem with the following properties is analyzed. The functional of the problem is bounded neither above nor below. There exists an extremal curve for this problem which is obtained by sewing together two different extremal curves and not twice differentiable at the sewing point. Despite this unfavorable situation, an attempt to apply the method of steepest descent (in the form proposed by V.F. Dem’yanov) to this problem is made. It turns out that the method converges to a stationary curve provided that a suitable step size rule is chosen.