Euler-Lagrange Equation question

Question

I learned from our textbook that Euler-Lagrange equation could be used to extremize a functional. But how can we tell that it maximize or minimize some given functional $F(y)$? It seems like we can't tell whether the result is a maximum or minimum

Answer 1

Just similar as what we learned in Calculus course, Euler-Lagrange equation plays the same role as the first derivative of a real-valued function. So what you obtain from Euler-Lagrange equation is like the "critical point" of the functional. To check it is the max./min., you need more information like second derivative, i.e. the second variation of the functional. And there are some criteria can help you check whether they are exactly max./min. or not. Maybe you can read some book about Calculus of variation for more detail.