This short textbook can quickly give you a good picture of what the calculus of variations (CoV) is about. But it may be best for those who have already encountered the subject in another context (e.g. a physics course), rather than for learning it from scratch.
I first heard of CoV in a classical mechanics course based on Goldstein in the '70s. While I got the gist of the specific application in that case, the idea of CoV always remained rather murky in my mind. With all due respect to the famous book by Lanczos, the only supplementary resource available back then, it was too long, wordy and philosophical to be attractive to a very average student like me. One of the great virtues of the present slim book is that it immediately gives you a very concrete understanding of the goals of CoV -- minus the metaphysical justifications for the topic that have clung to it since the times of Huygens and Maupertuis.
Like the applied math textbooks of my college days, this book focuses on calculation, rather than proof. But also like them, applications are discussed only rarely, at best. E.g., Hamilton-Jacoby theory and most other applications to mechanics aren't discussed at all. The exposition also usually assumes a Cartesian coordinate system, and other aspects of the notation will seem quaint (or annoying) to many readers who started reading textbooks in the 1990s or later. Some topics often discussed in modern textbooks, such as "second variation" to discriminate between weak and strong extrema, actually are included here but without using the modern terminology. Other topics, such as the Pontryagin maximum principle, are outside the scope of the book, and in fact the word optimization barely or perhaps never appears.
This 1961 book is also a good example of the clear Soviet pedagogy of a half century ago. However, the author takes it for granted that you will have a command over differential equations and analytical geometry typical for Soviet science and engineering students of the Space Age era -- which is a rather high bar to reach. There is also a bit of Cold War charm, as when the author announces that a certain equation "is called the Ostrogradski equation, after the famous Russian mathematician M.B. Ostrogradski, who discovered it first in 1834. It is also *sometimes* known as the Euler-Lagrange equation." (@50-51; my emphasis.)
I should mention one important caveat, concerning the discussion of Lagrange multipliers in Chapter 4: the exposition of this topic is quite brief and bears little resemblance to US pedagogy from around the same time as this book, much less the way it's presented now. The one or two examples applying the technique are also quite sketchy. I think you would have to be a *very* gifted student to develop a command over this topic for the first time -- or simply to learn much about it at all -- from this book. (Economists, especially, take note.) All in all, though, a great and inexpensive guide for those perplexed about the big picture of CoV.