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From Bernoulli’s brachistochrone to the concept of calibration. A historical outline on the rise of field theory in the calculus of variations (sufficient conditions in variational calculus). (Von der Bernoullischen Brachistochrone zum Kalibrator-Konzept. Ein historischer Abrißzur Entstehung der Feldtheorie in der Variationsrechnung (hinreichende Bedingungen in der Variationsrechnung).) (German) Zbl 1170.01006

De Diversis Artibus 80 (NS 43). Turnhout: Brepols Publishers (ISBN 978-2-503-52666-9; 978-2-503-52438-2). 828 p. (2007).
This is a book of History of Mathematics, not of Calculus of variations. The author discusses in detail the work of the Bernoullis, in particular the geometry based approaches, and then Carathéodory’s final synthesis of methods going back to Huyghens, Johann Bernoulli, Hamilton, and Jacobi. This is followed by a very detailed study of the work of Weierstrass though existing Notes from his many lectures on the Calculus of variations, followed by a study of H. A. Schwarz’s work and all other theses written under Weierstrass’s direction which made his work known to the world.
After a shorter chapter about fields of extremals, mainly in the works of Hamilton, Jacobi, and Kneser, another long chapter discusses in meticulous detail the work of Hilbert and his students, as well as that of Adolf Mayer. The next chapter is dedicated to later contribution to the notion of field of extremals, in particular also to the work of Osgood and the Chicago School (Bolza and Bliss), and others.
The last chapter gives a short survey of the work on problems of sufficient conditions and the appropriate definitions of multidimensional fields and appropriate transversality conditions in the middle of the past century. An appendix reproduces the author’s Habilitationsvortrag in Hamburg, 2002. The last 95 pages are documentation, bibliography, and indices.

MSC:

01A50 History of mathematics in the 18th century
01-02 Research exposition (monographs, survey articles) pertaining to history and biography
49-03 History of calculus of variations and optimal control
49K05 Optimality conditions for free problems in one independent variable
70-03 History of mechanics of particles and systems
70H25 Hamilton’s principle
70H30 Other variational principles in mechanics
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
49K99 Optimality conditions
49L99 Hamilton-Jacobi theories
70H20 Hamilton-Jacobi equations in mechanics
53C38 Calibrations and calibrated geometries
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