On Some Contributions to Field Theory in the Calculus of Variations from Beltrami to Caratheodory

Author: Thiele R.

Source: Historia Mathematica, Volume 24, Number 3, August 1997 , pp. 281-300(19)

Publisher: Academic Press

Abstract:

Hilbert's invariant integral, a prominent result in the calculus of variations, is associated with problem 23 of his famous lecture, "Mathematical problems," given at the International Mathematical Congress in Paris in 1900. Although Beltrami's investigation of non-Euclidean geometry in 1868 overshadowed his work that same year in the calculus of variations, his work in the latter area may be viewed as a precursor of Hilbert's ideas. The present paper traces Beltrami's and Hilbert's approaches to field theory. Adolf Kneser's concept of transversality linking the field lines (extremals) with the level surfaces of the eikonal provides a tool central to the analysis. Beltrami and Hilbert traveled the same mathematical route, but in opposite directions. In the long run, Caratheodory found the new approach to field theory, sometime called the "royal road."

Uno dei risultati piu rilevanti nel calcolo delle variazioni e noto come l'invariante integrale di Hilbert, ed e associato al ventitreesimo dei "Problemi Matematici" che Hilbert espose nel 1900 a Parigi durante il Congresso Internazionale di Matematica. Le ricerche di Beltrami del 1868 relative alla geometria noneuclidea, la sciarono in ombra il suo lavoro dello stesso anno sul calcolo delle variazioni. Possiamo tuttavia considerare Beltrami uno dei precursori di queste idee. Questo lavoro delinea i due approcci di Beltrami e di Hilbert alla teoria dei campi. Uno strumento centrale di confronto e rappresentato dal concetto di transversalita di Adolf Kneser che determina la connessione tra le linee di campo (estremali) e le superfici di livello della sua iconale. Beltrami e Hilbert viaggiano sullo stesso binario, ma in direzioni opposte. Alla fine fu Caratheodory a trovare un nuovo approccio, che viene talvolta indicato come la via regia verso la teoria dei campi.

Zu den bemerkenswerten Ergebnissen der Variationsrechnung zahlt das Hilbertsche Unabhangigkeitsintegral, das mit dem 23. Problem aus Hilberts beruhmten Vortrag "Mathematische Probleme," den er auf dem Internationalen Mathematiker-Kongres 1900 in Paris gehalten hat, verbunden ist. Die Beitrage Beltramis zur nichteuklidischen Geometrie von 1868 uberschatten dessen einschlagigen Arbeiten zur Variationsrechnung aus dem gleichen Jahr. Beltrami kann jedoch als Vorlaufer fur die von Hilbert ausgefuhrten Ideen angesehen werden. Die vorliegende Arbeit verfolgt sowohl Beltramis als auch Hilberts Zugang zur Feldtheorie. Als ein zentrales Hilfsmittel fur den Vergleich erweist sich dabei der Transversalitatsbegriff von Adolf Kneser, der den Zusammenhang zwischen den Feldkurven (Extremalen) und den Niveauflachen des Eikonals herstellt. Beltrami und Hilbert benutzen zwar den gleichen Weg, aber in gegensatzlichen Richtungen. Caratheodory war es schlieslich, der einen neuen Zugang, mitunter als Konigsweg bezeichnet, zur Feldtheorie fand.

Keywords: calculus of variations; field theory; geodesic field; strong extrema; invariant integral; transversality

Language: English

Document Type: Research article

Affiliations: Karl-Sudhoff-Institut, Universitat Leipzig, Augustusplatz 10-11, Leipzig, D-04109, Germany

Publication date: 1997-08-01

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