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.