What is a good book on the Calculus of Variations, for a second year PhD student?
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!
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.
A famous (and remarkable) text is by L C Young, lectures on the calculus of variations and optimal control theory, MR0259704.
I found this writing very intuitive and step-by-step exposition to easily understand the basic concepts. Thanks to Prof. Arnold Arthurs.