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

Hans Sagan

A chapter in Selecta Mathematica, 2002, pp 369-376 from Springer

Abstract: 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.

Date: 2002
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DOI: 10.1007/978-3-7091-6110-4_23

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