Why do we take just the first variation?

Question

Why, in calculus of variations, do we take just the first variation of the functional and never after that?

Are we just approximating or there is a reason that we don't have to take the second variation?

Answer 1

Second variation can be taken when appropriate. For example, Chapter 5 of Calculus of Variations by Gelfand and Fomin is about second variation, and it's over 30 pages. It may be just that you don't encounter it in the course of studying variational calculus, because

1. The concept is harder to grasp (bilinear form vs linear functional)
2. Computing the second variation is hard.
3. When it indicates an extremum, it's only a local extremum that we get.
4. There is only so much time in the course.