Introductory note to 1904a

pp. 494-511

in: Ernst Zermelo, Calculus of variations, applied mathematics, and physics/Variationsrechnung, angewandte mathematik und physik, Berlin, Springer, 2013

Abstract

Paul du Bois-Reymond (1879a,b) derived necessary conditions for only continuously differentiable extremals of J(y) = ∫ _a ^b f(x, y(x), y"(x)) dx → extr. and was able to show that with this weakening of the conditions, there are no additional solutions. In 1904a, Zermelo generalizes du Bois-Reymond's result to higher derivatives. He gives two proofs. The second proof is close to that of du Bois-Reymond, which considers an isoperimetric variational problem, and is again presented with an altered conclusion based on an observation by Erhard Schmidt.