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 [Formula: see text] 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.