THE CONSTRUCTION OF THE COMPLETE FIGURE IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS: A DIRECT APPROACH

Abstract

The field-theoretic approach to the calculus of variations of multiple integrals, as initiated by Carathéodory, is based on the systematic use of certain field components that exist if and only if a determinant whose entries are defined uniquely in terms of the Lagrangian and its derivatives does not vanish. If one does not introduce an assumption of this kind ab initio, entirely different techniques must be adopted. It is shown that these give rise to a surprisingly simple and direct construction of the complete figure by means of which the solutions of the standard variational problem may be realized.