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What is a good book on the Calculus of Variations, for a second year PhD student?

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    $\begingroup$ It might help to know what sort of research you are interested in. If you want to study applied mathematics you will probably have very different taste in the approach to the calculus of variations than a student of differential geometry. $\endgroup$
    – Ben McKay
    Commented Apr 7, 2019 at 13:20

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The book by Gelfand and Fomin is quite good (and its Dover ...). Another one I like a great deal are those of Giaquinta and Hildebrandt (specially volume 1), but those are not Dover: check them out from the library!

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    $\begingroup$ Surely Giaquinta&Hildebrandts represents the Bible, but there are more slender yet excellent references aroud. E.g., I have found very pleasant Bruce van Brunt's (amazon.com/Calculus-Variations-Universitext-Bruce-Brunt/dp/…), since "Enlightening explanations and building sound heuristics and intuition based on carefully chosen (classical) examples and exercises; simplicity and clarity of the exposition. But there is a price to pay: The mathematics is approached rigorously, but the level of rigor and details may not satisfy the purists among the mathematicians." [MR review] $\endgroup$
    – Giovanni Moreno
    Commented Nov 26, 2015 at 15:04
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The book of Gelfand and Fomin is a good place to start. It worked for me. I would like to include another nice and short source namely Chapter 19, vol. II of Feynman's Lectures on Physics.

If you know a little about smooth manifolds, then Arnolds's Mathematical Methods of Classical Mechanics is another excellent source. Also, check volume 1 of Dubrovin, Fomenko, Bovikov, Modern Geometry.

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    $\begingroup$ I want to second the recommendation on Feynaman's book, as it gives the analytical-mechanics intro to calculus of variation. As far as I'm concerned, it is the natural way to motivate and justify the approach as a whole. $\endgroup$
    – Amir Sagiv
    Commented May 13, 2016 at 6:58
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A famous (and remarkable) text is by L C Young, lectures on the calculus of variations and optimal control theory, MR0259704.

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    $\begingroup$ It is indeed a remarkable text. I think than Young measures were introduced there. The book is even worth reading only for its jokes and anecdotes! Let me also add Caratheodory's Calculus of Variations and Partial Differential Equations of First Order. $\endgroup$
    – alvarezpaiva
    Commented Apr 29, 2013 at 20:33
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I found this writing very intuitive and step-by-step exposition to easily understand the basic concepts. Thanks to Prof. Arnold Arthurs.

http://www.math.unipd.it/~taylor/files/york/CalculusofVariations.pdf

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    $\begingroup$ Prof Arthurs taught me Variational Calculus in the Autumn term 2000. These notes formed the basis of that course, I have to say that was one of the greatest lecture courses I studied in my MMath. $\endgroup$
    – asymptotic
    Commented Dec 12, 2016 at 14:40
  • $\begingroup$ Hi: Would anyone happen to know if the pdf still exists somewhere because the link above no longer works ? Thanks. $\endgroup$
    – mark leeds
    Commented Aug 21, 2021 at 5:23
  • $\begingroup$ @markleeds: I found it here: naderpour.semnan.ac.ir/uploads/8/2020/Mar/13/… and here: academia.edu/31578059/CALCULUS_OF_VARIATIONS $\endgroup$
    – kjetil b halvorsen
    Commented Jan 1, 2022 at 19:33
  • $\begingroup$ thanks for finding it. $\endgroup$
    – mark leeds
    Commented Jan 2, 2022 at 20:03

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