Composition functionals in calculus of variations.: Application to products and quotients

This paper deals with the problem of the Calculus of Variations for a functional which is the composition of a certain scalar function H with the integral of a vector valued field f, i.e. of the form H (integral(x1)(x0) f(x, y(x), y'(x)) dx), where H : R-n -> R and f : R-3 -> R-n. The integral of f is calculated here componentwise. We examine sufficient conditions for the existence of optimal solutions, and provide rules to obtain the necessary Euler-Lagrange, natural, transversality, Weierstrass-Erdmann and junction conditions for such a functional. Particular attention is paid to the cases of the product and the quotient as we take these as model situations. Finally, the theory is illustrated with a slope stability problem, and an example coming from Economics.