Can someone please tell me what the best book for learning calculus of variations is?
At what level, for what purposes? The physical, computational way, or the mathematically rigorous way?
I'd prefer the mathematically rigorous way. I first encountered calculus of variations in my graduate mechanics class, and we did a few problems with it, but I never really understood it completely. (I understand that it's one way to derive the Euler-Lagrange equations.)
Is there a text, adequate for self-study, that lays out the rigorous mathematical framework and then goes on to apply the theory to physical problems, like deriving the Euler-Lagrange equations or showing that the shortest path between two points in the plane is a straight line?
For the computational approach I would say Goldstein has a pretty clear explanation.
https://www.amazon.com/Calculus-Variations-I-M-Gelfand/dp/0486414485 is a great classic text (Dover, cheap), see Google books to browse through it. It is theoretical, but with a lot of physics applications (and a clear lay out of Noethers theorem, which I couldn't really follow in one of my physics classes).
A more modern book is https://www.amazon.com/Calculus-Var...r_1_12?ie=UTF8&s=books&qid=1261346582&sr=1-12 by Jürgen Jost and another Li-Jost. This one goes deeper, using functional analysis and measure theory in the second part.
Then there's another https://www.amazon.com/Calculus-Variations-Universitext-Bruce-Brunt/dp/0387402470 (not very original names) which seems ok, but I haven't read this one.
Tray B. Dacorogna:Introduction to the Calculus of Variations (Paperback)
Paperback: 300 pages
Publisher: Imperial College Press; 2 edition (December 10, 2008)
I learned to love the subject from Gelfand and Fomin.
Yes, Gelfand & Fomin , a fine classic. Very nice. K.
Separate names with a comma.