Calculus of Variations/PREFACE

Since the time of Newton and the Bernoullis, problems have been solved by methods to which the general name of the Calculus of Variations has been applied. These methods were generalized and systematized by Euler, Lagrange, Legendre and their followers; but numerous difficulties arose. Some of these were removed by Jacobi and his contemporaries. Still many of the methods had to be extended and it was necessary to supply much that was deficient and to make clear what remained obscure. The progress of Analysis is indebted to the genius of Weierstrass for the perfection of this theory.

While a student in the University of Berlin it was my privilege to hear the lectures of Professor H. A. Schwarz on the Calculus of Variations. In its presentation this eminent mathematician followed his great teacher, Weierstrass, who had established the theory on a firm foundation, free from objection, simple and at the same time more comprehensive than it had been hitherto. I also took the opportunity to study Weierstrass's lectures, of which there were copies in the Mathematischer Verein.

Through the courtesy of Professor Ormond Stone abstracts of this theory were published in Volumes IX, X, XI and XII of the Annals of Mathematics and from time to time the Calculus of Variations has been included in my University lecture courses.

I have delayed the publication of these lectures with the hope that Weierstrass's lectures would be published by the commission to which has been intrusted the editing of his complete works. This publication, however, seems remote, and commentaries on the Calculus of Variations are becoming so numerous that I have deemed it expedient to bring out my work at the present time.

As one would naturally expect, I have followed Weierstrass's treatment of the subject; in many places, especially in the latter part of the book, my lectures are little more than a repetition of his. It is from the Weierstrassian standpoint that I have developed my own ideas and have presented those derived from other writers. Thus, instead of giving separate accounts of Legendre's and Jacobi's works introductory to the general treatment, I have produced their discoveries in the proper places in the text, and I believe that by this means confusion has been avoided which otherwise might be experienced by students who are reading the subject for the first time. I hope that this exposition of the fundamental principles may prove attractive. The reader will then naturally desire a more extensive knowledge regarding the literature and the various improvements that have been made by successive mathematicians. He will wish to follow the methods which they have employed and will seek further information regarding the historical development. References are given on Pages 18 and 19 of the text from which the original sources are easily obtained.

The necessary and sufficient conditions as they arise for the existence of a maximum or a minimum are illustrated by six problems, which are worked out step by step in the theory. They have been chosen to represent the different phases of the subject, the exceptional cases which may occur, the discontinuous solutions, etc. For example, in the first problem it is found that a minimum may be offered by an irregular curve, whereas seemingly the problem is satisfied by a regular curve, the catenary. Attention is thereby called to the fact that although our integrals have a meaning only when taken over regular curves, we have to guard against discontinuous solutions, and consequently further conditions for the existence of a maximum or a minimum must be derived. The case of the discontinuous solution is considered in this problem as also when the limits of integration are two conjugate points. Newton's problem is introduced to show that one of the necessary conditions is not satisfied and that there is no curve which fulfills the given requirements.

By the formulation of such problems in Chapter I we come readily to the statement of the general problem of the Calculus of Variations.

In the general discussion attention has been confined for the most part to the realm of two variables, and in this realm only the first derivatives of the variables have been admitted. Generalizations and extensions are suggested which, as a rule, may be executed with little difficulty.

The second part of the work beginning with Chapter XIII treats of the theory of Relative Maxima and Minima, where the isoperimetrical problems are considered. Here also the existence of a field about the curve which is to maximize or minimize a given integral is emphasized and the necessary and sufficient conditions are derived and proved in a manner similar to that by which the analogous conditions are found in the first part of the work.

My wish in these lectures has been to give a connected and simple treatment of what may be called the Weierstrassian Theory of the Calculus of Variations. Many instructive theorems of older writers have been omitted. I regret too that there has not been room to take up some of the investigations which have recently appeared. It is seldom that the first edition of a book is the final form in which an author wishes to leave his work. As I expect to make additions and alterations in my University lectures from time to time, I shall receive with pleasure any suggestions that ma be offered.

In conclusion, I wish to take the opportunity here of returning my sincere thanks to the Board of Directors of the University of Cincinnati for their liberality in the publication of this work.

My thanks are also due to Mr. Harold P. Murray, Manager of the University Press, for his careful supervision of the printing.

Harris Hancock.

Auburn Hotel,
 * Cincinnati, O.
 * April 15, 1904.