LECTURES ON THE CALCULUS OF VARIATIONS OSKAK BOLZA OF THE DJSPARTMKNT OF M VTHKM VTKS THE DECENNIAL PUBLICATIONS SECOND SERIES VOLUME XIV CHICAGO THE UNIVERSITY OF CHICAGO PRESS 1904 OF 6F BY TllK UNIVKRK1TY OK CHKAUO September, 1904 PEEFACE THE principal steps in tlie progress of the Calculus of Variations during the last thirty years may be characterized as follows 1. A critical revision of the foundations and demonstra tions of the older theory of the first and second variation according to the modern requirements of ...
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LECTURES ON THE CALCULUS OF VARIATIONS OSKAK BOLZA OF THE DJSPARTMKNT OF M VTHKM VTKS THE DECENNIAL PUBLICATIONS SECOND SERIES VOLUME XIV CHICAGO THE UNIVERSITY OF CHICAGO PRESS 1904 OF 6F BY TllK UNIVKRK1TY OK CHKAUO September, 1904 PEEFACE THE principal steps in tlie progress of the Calculus of Variations during the last thirty years may be characterized as follows 1. A critical revision of the foundations and demonstra tions of the older theory of the first and second variation according to the modern requirements of rigor, by WEIER STBASM, ERDMANN, Du BOIS-REYMOND, SCHEEFEER, SCHWARZ, and others. The result of this revision was a sharper for mulation of the problems, rigorous proofs for the first three necessary conditions, and a rigorous proof of the sufficiency of those conditions for what is now called a weak exte rn urn. 2. WEIERSTBASSS extension of the theory of the first and second variation to the case where the curves under consid eration, are given in parameter-representation. This was an advance of great importance for all geometrical applications of the Calculus of Variations for the older method implied for geometrical problems a rather artificial restriction. 8. WEIERSTRASSS discovery of the fourth necessary con dition, and his sufficiency proof for a so-called strong extrernum, which gave for the first time a complete solution, at least for the simplest typ of problems, by means of an entirely new method based upon what is now known as WEIERSTBASSS construction. These discoveries mark a turning-point in the history of the Calculus of Variations. Unfortunately they were given by WBIEBSTBASS only in his lectures, and thus became known only very slowly to the generalmathematical public. Chiefly under the influence of WEIERSTBASSS theory a vigorous activity in the Calculus of Variations has set in PREFACE during the last few years, which has led-apart from exten sions and simplifications of WEIEKSTRASSS theory to the following two essentially new developments 4. KNESEKS theory, which is based upon an extension of certain theorems on geodesies to extremals in general. This new method furnishes likewise a complete system of suffi cient conditions and goes beyond WEIEKSTRASSS theory, inasmuch as it covers also the case of variable end - points. 5. HILBERTS a priori existence proof for an extremum of a definite integral a discovery of far-reaching impor tance, not only for the Calculus of Variations, but also for the theory of differential equations and the theory of functions. To give a detailed account of this development was the object of a series of lectures which I delivered at the Collo quium held in connection with the summer meeting of the American Mathematical Society at Ithaca, N. Y., in August, 1901. And the present volume is, in substance, a reproduc tion of these lectures, with such additions and modifications as seemed to me desirable in order that the book could serve as a treatise on that part of the Calculus of Variations to which the discussion is here confined, viz., the case in which the function under the integral sign depends upon a plane curve and involves no higher derivatives than the first. With this view I have throughout supplied the detail argu mentation and introduced examples in illustration of the gen eral principles. The emphasis lies entirely on the theoretical side I have endeavored to give clear definitions of the fundamental concepts, sharp formulations of the problems, and rigorous demonstrations. Difficult points, such as the proof of the existence of a field, the details in HILBERTS exist ence proof, etc., have received special attention. For a rigorous treatment of the Calculus of Variations the principal theorems of the modern theory of functions of a real variable are indispensable these I had therefore to PREFACE xi presuppose, the more so as I deviate from WEIERSTRASS and KNBSEE in not assuming the function under the integral sign to be analytic...
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