Introductory note to 1902d

pp. 484-493

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

Abstract

Zermelo considers relative and absolute minima of geodesics, which he calls shortest and by far shortest paths, respectively. The infinitesimal variational technique first takes into account only sufficiently close comparison functions that lie in a given neighborhood of the extremal, leading to necessary conditions for relative extrema, in this case minima. Zermelo mentions three possible ways to extend the associated variational problem, which can be characterized by the following key words: (a) absolute minima, (b) restrictions on surfaces, (c) differential inequalities as constraints. The cases (b) and (c) appear naturally in practical questions, among them the problem of road and rail construction.