What is the best book for learning calculus of variations?

[AxiomOfChoice]

Can someone please tell me what the best book for learning calculus of variations is?

 

[Landau]

At what level, for what purposes? The physical, computational way, or the mathematically rigorous way?

 

[AxiomOfChoice]

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?

 

[Landau]

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.)

For the computational approach I would say Goldstein has a pretty clear explanation.

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?

https://www.amazon.com/dp/0486414485/?tag=pfamazon01-20 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/dp/0521057124/?tag=pfamazon01-20 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/dp/0387402470/?tag=pfamazon01-20 (not very original names) which seems ok, but I haven't read this one.

 

[kowalski]

Tray B. Dacorogna:Introduction to the Calculus of Variations (Paperback)

Paperback: 300 pages
Publisher: Imperial College Press; 2 edition (December 10, 2008)
Language: English
ISBN-10: 1848163347
ISBN-13: 978-1848163348

Kowalski

 

[Cantab Morgan]

I learned to love the subject from Gelfand and Fomin.

 

[kowalski]

I learned to love the subject from Gelfand and Fomin.

Yes, Gelfand & Fomin , a fine classic. Very nice. K.