Introductory text for calculus of variations

Solution 1:

If you check out Wikipedia's entry on "Calculus of Variations: here, and scroll down to the bottom where "References" are listed:

You'll find a link to a pdf reference (Jon Fischer, Introduction to the Calculus of Variation, a quick and readable guide) that might be exactly what you're looking for, as well as some additional references (sample problems, guides, etc.).
In addition, you'll find a link to this site listed among the references.
There's also a chapter of a text that's available online: Chapter 8: Calculus of Variation from Optimization for Engineering Systems, by Ralph W. Pike.

There are also some additional texts and resources listed in the linked Wikipedia's entry, as well.

Solution 2:

I just purchased a copy of Gelfand and Fomin's Calculus of Variations. It's a Dover book, so it's really inexpensive (I paid $9, including shipping).

The book gets very good reviews, both on Amazon and MathOverflow. Just from reading the first few pages, it looks quite promising.

Solution 3:

I know this post is old, but if anyone else is looking for a good, concise and intuitive introduction to the calculus of variations, the chapter 'calculus of variations' in Peter Olver's as yet unpublished 'Applied Mathematics' (well, the first 10 chapters are published as 'Applied Linear Algebra') is very readable.

As of September 2011, this chapter is available on Peter's website at http://www.math.umn.edu/~olver/appl.html

Solution 4:

Michael Struwe Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Fourth Edition, Springer, 2008.

Solution 5:

One I really enjoy is Calculus of Variations by Jürgen Jost - he also has an awesome book on Partial Differential Equations!