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

$\begingroup$ 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 Apr 7 at 13:20
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!

1$\begingroup$ 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/CalculusVariationsUniversitextBruceBrunt/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 Nov 26 '15 at 15:04
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.

$\begingroup$ I want to second the recommendation on Feynaman's book, as it gives the analyticalmechanics 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 May 13 '16 at 6:58
A famous (and remarkable) text is by L C Young, lectures on the calculus of variations and optimal control theory, MR0259704.

1$\begingroup$ 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 Apr 29 '13 at 20:33
I found this writing very intuitive and stepbystep exposition to easily understand the basic concepts. Thanks to Prof. Arnold Arthurs.
http://www.math.unipd.it/~taylor/files/york/CalculusofVariations.pdf

$\begingroup$ 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. $\endgroup$ – Kevin Dec 12 '16 at 14:40