A Mengerian Tour Along Caratheodory’s Royal Road

Abstract

It is ironic that Caratheodory’s work on the foundations of thermodynamics is better known today [1, 2] than his equally elegant formulation of the calculus of variations, once called “Königsweg” by specialists. This formulation is set forth in Chapters 12 and 13 of Caratheodory’s book [3] and also discussed in detail in [4]. The seminal idea for this development is Hilbert’s invariant integral (which in [5] is likened to the gauge transformation of electrodynamics), from which perspective the calculus of variations is so formulated that the Hamilton-Jacobi theory and Hamilton’s canonical equations are developed in parallel with the Beltrami and Euler equations. Another striking feature is that sufficient conditions for extremization arise naturally and immediately, in contrast with the historical development, following Jacobi and Weierstrass, in which sufficiency is achieved much less directly [6].