Peter Olver from U. Minnesota writes some of the best tutorial notes available and they are free on his website. He has a great 27 page packet on introducing Calculus of Variations and it is worth your time to take a look:

http://math.umn.edu/~olver/ln_/cv.pdf

Andreas I will suggest you those links: 

The Calculus of Variations

http://math.hunter.cuny.edu/mbenders/cofv.pdf

TOPICS ON FUNCTIONAL ANALYSIS, CALCULUS OF VARIATIONS AND DUALITY

Fabio Botelho

Hi, thank you very much for the quick answer! I'll make sure to look into the resources you suggested over this weekend.

A colleague yesterday recommended Francis Clarke's book (Functional Analysis, Calculus of Variations, and Optimal Control). I only had time to browse through it since yesterday, but it seems good.

Peter Olver from U. Minnesota writes some of the best tutorial notes available and they are free on his website. He has a great 27 page packet on introducing Calculus of Variations and it is worth your time to take a look:

http://math.umn.edu/~olver/ln_/cv.pdf

Try these books:

• Calculus of Variations (Dover Books on Mathematics) Paperback – October 16, 2000

by I. M. Gelfand (Author), S. V. Fomin (Author)

▪ The Calculus of Variations (Universitext)

Bruce van Brunt

Following classical books have excellent treatment on Calculus of variations::

1.  Methods of Mathematical physics - vol. I by Courant and Hilbert (Ch. IV)

2. Methods of Applied Mathematics - Hildebrand

To complement Michael Wendl's answer, after you gain some basic knowledge of calculus of variations you may also wish to look into Peter Olver's book Applications of Lie Groups to Differential Equations (https://www.researchgate.net/publication/200175449_Applications_of_Lie_Groups_to_Differential_Equations ), especially into Chapters 4 and 5 discussing symmetries in the calculus of variations and the celebrated Noether theorem.

Book Applications of Lie groups to differential equations. 2nd ed

A good book is: The Calculus of Variations by Bruce  van Brunt.

I suggest the following books, with modern notation, direct methods and discussion on the existence and the regularity theory:

Giusti Enrico, Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003. viii+403 pp. ISBN: 981-238-043-4

Dacorogna Bernard, Direct methods in the calculus of variations. Applied Mathematical Sciences, 78. Springer-Verlag, Berlin, 1989. x+308 pp. ISBN: 3-540-50491-5, with a second edition in 2008.

Try this one: "Moser, Jürgen Selected chapters in the calculus of variations." It is short, but have good ideas and explanations.

I found this writing very intuitive and step-by-step exposition to easily understand the basic concepts. Thanks to Prof. Arnold Arthurs.

http://www.math.unipd.it/~taylor/PDF/York/CalculusofVariations.pdf

Thank you! It sure has a welcoming size for a primer on the subject.