# Introductory text for calculus of variations

I am currently working on problems that require familiarity with calculus of variations. I am fairly new to this field. Please suggest a good introductory book for the same that could help me pick up the concepts quickly.

edit: I would prefer books which are available in PDF format online.

• Do you have access to a library? Sometimes they have books available digitally that might otherwise be unavailable (legally). Jun 11, 2011 at 20:01
• @Nick I do have access to the university library. Jun 11, 2011 at 22:18

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

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

• The links in your 2nd and third bullet points are dead. Feb 24, 2018 at 18:58
• The paper in the first link, albeit good, needs some edits... e.g. the diffeq in Theorem 1 — the most important Theorem — is wrong! Mar 31, 2018 at 16:47
• @wcochran it would be great if you could also share what's wrong. Thanks! Jun 20, 2021 at 23:42
• The diff eq at the bottom of page 5 should be \partial f / \partial y - d/dx (\partial f / \partial y') = 0. I am sure it's just a typo, but kind of important to get the main equation right. Jun 22, 2021 at 2:41

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.

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and intuitive introduction to the calculus of variations, the chapter 'calculus of variations' in Peter Olver's as yet unpublished 'Applied Mathematics' (well, the first 10 chapters are published as 'Applied Linear Algebra') is very readable.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

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

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

Charles MacCluer wrote a book on the subject in 2008 for students with a minimal background (basically calculus and some differential equations), Calculus of Variations: Mechanics, Control and Other Applications. I haven't seen the whole book,but what I have seen is excellent and very readable. MacCluer says in the introduction his goal was to write a book on the subject that doesn't replace the old classics, but updates and supplements them with a lot of real-world applications and without heavy prerequisites. From what I've seen, he's succeeded and best of all, the book's available from Dover in a cheap paperback. This might be just what you're looking for.

If you're looking for something more mathematicially sophisticated and up-to-date, you can try the book by Bruce Van Brunt.