Calculus of Variations (Dover Books on Mathematics) by Lev D. Elsgolc (PDF)

Description

This concise text offers both professionals and students an introduction to the fundamentals and standard methods of the calculus of variations. In addition to surveys of problems with fixed and movable boundaries, it explores highly practical direct methods for the solution of variational problems.Topics include the method of variation in problems with fixed boundaries; variational problems with movable boundaries and other problems; sufficiency conditions for an extremum; variational problems of constrained extrema; and direct methods of solving variational problems. Each chapter features numerous illustrative problems, and solutions appear at the end.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I used to think that the best introduction to Calculus of Variations for scientists was Weinstock’s book

⭐. I had discovered it as an undergrad student and, since then, I kept it at the top of my list. But, recently, I discovered Elsgolc’s book and I must admit that it has to replace Weinstock’s book at the top of the list.Weinstock’s book is written at the right level for scientists – not highly abstract, but not imprecise for mathematicians. It contains all the right topics from calculus of variations and lots – really lots – of applications from science. However, I always felt that all these applications are more of distraction for someone who wants to learn just the topic of calculus of variations and then apply it in his/her own discipline. After all, the majority of the applications material in Weinstock’s book can be found in physics and engineering books easily; and in these applications the calculus of variations part is only a small step to get a differential equation for the phenomenon under consideration. But the actual theory of the calculus of variations cannot be found so easily in the science books. Usually, these books devote a brief chapter to the topic of calculus of variations discussing only the main problem (which is often solved in a very unsatisfying way) and then state that other problems can be dealt similarly, essentially asking the reader to discover the remaining techniques on his/her own.My point in the previous paragraph is that scientists are in need to read a presentation of the various calculus of variations techniques in a crystal clear way, not read a copy of their mechanics text. And Elsgolc’s book is exactly this: a careful, clean presentation of the theory without extremely long and unnecessary excursions to physics. It is written at the same level as Weinstock’s book and it does contain simple examples to clarify the theory. One nice feature of the book is many two-column pages in which the author shows to the reader how the ideas of calculus of variations are similar to the ideas of the traditional calculus (of functions). The beginners will find this feature very valuable. The book is thinner than Weinstock’s and yet it contains more topics than Weinstock’s: Weinstock does not discuss extremals with cusps, neither he deals with sufficiency conditions for an extremum. The book does contain some quick applications (section 7 of chapter 1) to make the connection with science. The reader will find Weinstock’s end-of-chapter problems more interesting and exciting but Elsgolc’s are not bad. They are made to fit exactly the contents of the corresponding chapter. Most probably, beginners will find these problems more useful just because they are fine-tuned to the material.Finally I should say that, although I have presented the book as a good reading for scientists, it is also a good reading for mathematicians. In fact, mathematicians will be turned off by Weinstock’s book. The extensive applications make the math scant. So, Elsgolc’s book is the right choice.Overall, if you want to learn the topic, I recommend this book as first reading.

⭐This short textbook can quickly give you a good picture of what the calculus of variations (CoV) is about. But it may be best for those who have already encountered the subject in another context (e.g. a physics course), rather than for learning it from scratch.I first heard of CoV in a classical mechanics course based on Goldstein in the ’70s. While I got the gist of the specific application in that case, the idea of CoV always remained rather murky in my mind. With all due respect to the famous book by Lanczos, the only supplementary resource available back then, it was too long, wordy and philosophical to be attractive to a very average student like me. One of the great virtues of the present slim book is that it immediately gives you a very concrete understanding of the goals of CoV — minus the metaphysical justifications for the topic that have clung to it since the times of Huygens and Maupertuis.Like the applied math textbooks of my college days, this book focuses on calculation, rather than proof. But also like them, applications are discussed only rarely, at best. E.g., Hamilton-Jacoby theory and most other applications to mechanics aren’t discussed at all. The exposition also usually assumes a Cartesian coordinate system, and other aspects of the notation will seem quaint (or annoying) to many readers who started reading textbooks in the 1990s or later. Some topics often discussed in modern textbooks, such as “second variation” to discriminate between weak and strong extrema, actually are included here but without using the modern terminology. Other topics, such as the Pontryagin maximum principle, are outside the scope of the book, and in fact the word optimization barely or perhaps never appears.This 1961 book is also a good example of the clear Soviet pedagogy of a half century ago. However, the author takes it for granted that you will have a command over differential equations and analytical geometry typical for Soviet science and engineering students of the Space Age era — which is a rather high bar to reach. There is also a bit of Cold War charm, as when the author announces that a certain equation “is called the Ostrogradski equation, after the famous Russian mathematician M.B. Ostrogradski, who discovered it first in 1834. It is also *sometimes* known as the Euler-Lagrange equation.” (@50-51; my emphasis.)I should mention one important caveat, concerning the discussion of Lagrange multipliers in Chapter 4: the exposition of this topic is quite brief and bears little resemblance to US pedagogy from around the same time as this book, much less the way it’s presented now. The one or two examples applying the technique are also quite sketchy. I think you would have to be a *very* gifted student to develop a command over this topic for the first time — or simply to learn much about it at all — from this book. (Economists, especially, take note.) All in all, though, a great and inexpensive guide for those perplexed about the big picture of CoV.

⭐I needed only a basic knowledge of the large amount this book provides about variational calculus.I had looked at online resources and you tube videos. But I was unable to understand how or why the Euler equation for ordinary functionals is developed the way it is. If too many steps were skipped, I got lost. This author approached the Euler equation by two methods. He explained exactly everything he did. For me, he did skip a little bit, but he explained enough detail that I could fill in my blanks. Finally I got it, and could derive and solve basic equations.The style may be too detailed and verbose for somebody who has already mastered the subject. For someone who is trying to grasp the basic reasons for it, I would give this book a try. Dover Books picked another winner.

⭐The subject matter requires a good facility in calculus and some differential equations. Elsgolc does a magnificent job in presenting the topics in a clear and motivated way. A comprehensive catalog of possible variational problems – simple, multivariate- is developed. Problems elucidating the different items are worked out and explained. A good set of problems for the student are provided along with the answers and hints.Also, unlike most kindle editions, this one is satisfactory.

⭐Concise, clearly presented, and straight to the point. I found this book very helpful for strengthening my understanding of the calculus of variations.

⭐Very nice engineering book

⭐It was just terrible. I honestly don’t know how to put it … it felt like something was squeezing my brain whenever I read it for my class. I had to take breaks almost every page just to not fall asleep

⭐Purtroppo si tratta di una riproduzione per imagine e non una ristampa.Pertanto, anche se si tratta di un bellissimo libro di autore autorevole e chiaro, il testo risulta poco leggibile.

⭐

⭐This book is the bible for Calculus of Variations. I recommend this book along with Gelfand and Fomin’s book for those who want to have an introductory self reading course on the subject.

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