A Mengerian Tour Along Caratheodory’s Royal Road
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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  and also discussed in detail in . The seminal idea for this development is Hilbert’s invariant integral (which in  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 .
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- 2.W. G. V. Rosser: An Introduction to Statistical Physics, Halsted Press, New York, 1982.Google Scholar
- 3.C. Caratheodory: Variationsrechnung und partielle Differentialgleichungen erster Ordnung, Teubner, Leipzig and Berlin, 1935.Google Scholar
- 5.H. Rund: The Hamilton-Jacobi Theory in the Calculus of Variations, D. Van Nostrand, London, 1966.Google Scholar
- 6.H. Sagan: Introduction to the Calculus of Variations, McGraw-Hill, New York, 1969.Google Scholar