Commentary on Menger’s Work on the Calculus of Variation and Metric Geometry

Abstract

The simplest variational problem as it was set forth by Karl Weierstraß in his lectures commencing in 1872 is the following: To be found is a curve C: x _k=x _k(t), t ₁≤t≤ t ₂(k = 1, 2, 3,…n) joining two fixed points P and Q which renders the integral

$${{\lambda }_{\phi }}(C) = \int_{P}^{Q} {\phi (x,x\prime )dt}$$

(*)

a minimum. Here∅ is a given function, x stands for (x ₁(t), x ₂(t), x ₃(t),…, x _n(t)) and x’ stands for (x’₁ (t), x’₂(t), x’₃t),..., x’_n(t)). Everything that will be said about a minimum holds mutatis mutandis for a maximum.