Calculus of VariationsBased on a series of lectures given by I. M. Gelfand at Moscow State University, this book actually goes considerably beyond the material presented in the lectures. The aim is to give a treatment of the elements of the calculus of variations in a form both easily understandable and sufficiently modern. Considerable attention is devoted to physical applications of variational methods, e.g., canonical equations, variational principles of mechanics, and conservation laws. 
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great book, but don't get it here
This is a great book, but don't get it here. Google books forces me to read it on their online viewer, and I can't actually download it. This really sucks too because the online book viewer is bad and I need internet to access it.
very good book, explained the things i was looking for with details. its a pity in 122 it doesnt speak about lagrange factors though!
Contents
ELEMENTS OF THE THEORY  1 
FURTHER GENERALIZAT1ONS  34 
THE GENERAL VARIATION OF A FUNCTIONAL  54 
THE CANONICAL FORM OF THE EULER EQUATIONS AND RELATED TOPICS  67 
THE SECOND VARIATION SUFFICIENT CONDITIONS FOR A WEAK EXTREMUM  97 
FIELDS SUFFICIENT CONDITIONS FOR A STRONG EXTREMUM  131 
VARIATIONAL PROBLEMS INVOLVING MULTIPLE INTEGRALS  152 
DIRECT METHODS IN THE CALCULUS OF VARIATIONS  192 