CONDITIONAL EXTREMUM PROBLEMS OF FUNCTIONALS. ISOPERIMETRIC PROBLEMS AND SOME REMARKS ON THEM

Abstract

This paper investigates conditional extremum problems in variational calculus, with particular focus on the Lagrange problem and isoperimetric problems. The Lagrange problem considers the extremum of a functional depending on several unknown functions subject to both boundary conditions and additional differential constraints. Using the classical Lagrange multiplier method, the conditional extremum problem is reduced to an unconstrained extremum problem via the construction of an augmented Lagrange function. The necessary conditions for the extremum are formulated through the corresponding Euler–Lagrange equations. Isoperimetric problems, in which the additional constraints are given in integral form, are treated as a natural generalization of the Lagrange problem; it is shown that the two classes differ in that isoperimetric multipliers are real constants rather than functions. Theoretical results are illustrated through two worked examples: the first finds the extremal curves for a two-function Lagrange problem with a differential constraint, and the second solves an isoperimetric problem for a functional subject to an integral side condition. In both cases the extremal functions are obtained explicitly by solving the resulting systems of Euler equations together with the prescribed boundary and auxiliary conditions.