Integral formulae associated with the Euler-Lagrange operators of multiple integral problems in the calculus of variations

Abstract

All known field theories of m-fold integral variational problems are based on Carath6odory's method of equivalent integrals; the latter result from certain integrands which are such that the corresponding Euler-Lagrange equations are satisfied identically for all field functions. However, the integrands used for this purpose are not necessarily the only functions possessing this property, and accordingly an exhaustive characterization is given of all classes of integrands which cause the Euler-Lagrange equations to be satisfied identically. The functions thus obtained are useful only if they give rise to independent integrals, which is possible if and only if one may associate with them certain integral formulae which are non-trivial in that the resulting (m-1) fo ld integrals over the boundary aG of the m-dimensional domain of integration G assume the same values for all field functions which coincide on dG, even when the derivatives of these functions differ on t~G. It is shown that each integrand for which the Euler-Lagrange equations are identically satisfied does indeed give rise to an integral formula which is non-trivial in this sense. Moreover, it is found that this class of integrands contains those on which the known field theories are based as a proper subclass, and accordingly it is possible at least in theory to construct far more general field theories of multiple integral problems in the calculus of variations. This is not attempted here; instead, the integral formulae are used to derive sufficiency conditions for optimal control problems whose performance index is represented by an m-fold integral and whose state equation is defined by a system of first order partial differential equations.