J. C.  Clegg, Calculus of Variations (Oliver and Boyd, Edinburgh, 1968). An excellent elementary book with many solved problems and exercise hints.

I. M.  Gelfand and S. V.  Fomin, Calculus of Variations (Prentice-Hall, New Jersey, 1963). An excellent exposition in the classical style, including some discussion of mechanics and optimal control.

L. Pars, An Introduction to the Calculus of Variations (Heinemann, London, 1962). A delightful classic with many solved problems.

J. L.  Troutman, Variational Calculus with Elementary Convexity (Springer-Verlag, New York, 1983). A modern undergraduate text. An excellent presentation of the mathematical ideas behind classical optimization.

M. J.  Sewell, Maximum and Minimum Principles (Cambridge U.P., Cambridge, 1987). A modern undergraduate text with a very broad range of applications treated in detail.

H. Sagan, Introduction to the Calculus of Variations (McGraw-Hill, New York, 1969). This great book is unfortunately out of print, but it is in our library [QA 315 S23]. The style is modern and very precise with many details worked out. See also his excellent book on advanced calculus.

C. Carathéodory, Calculus of Variations and Partial Differential Equations of the First Order (Chelsea, New York, 1982). A re-issue of a grand old classic. If you like Sommerfeld for mechanics, then this is the corresponding grand master to look at for calculus of variations. It's not easy. The style is more concrete than many contemporary works; this feature is sometimes welcome and sometimes not.

K. Rektorys, Variational Methods in Mathematics, Science, and Engineering (Reidel, Dordrecht, 1980). In this amazing volume one has analysis, approximation theory, operator theory, and detailed studies of a vast collection of variational problems.

A. M.  Arthurs, Complementary Variational Principles (Oxford U.P., Oxford, 1980). A rather special work describing ways of finding both upper and lower bounds. See also Weinstein and Stenger. [Note: Arthurs also wrote a delightful short elementary book on calculus of variations.]

L. Pars, A Treatise on Analytical Mechanics (John Wiley, New York, 1965). The title describes this book very well: it is a masterful presentation of the subject with copious fascinating problems worked out in leisurely detail. Strongly recommended!

C. Lanczos, The Variational Principles of Mechanics (University of Toronto Press, Toronto, 1970). This is mechanics set to music. It is a grand view of the mathematical and physical ideas.

Robert  Rosen, Optimality principles in biology (Butterworth, London, 1967). This wonderful little book is unfortunately out of print, but it is in our library [QH 324 R67]. Related titles can by found by searching under theoretical biology.