I may have the wrong group. I could not find calculus of variations and had to start somewhere.
In the calculus of variations we start by finding the 0 points where the functions are at minimum or maximum.
Is this the same as stationary that is referred to in standard textbooks ?
i.e where the integral = I[f] is stationary. Can stationary be replaced with 0? I get confused when a new word is added to describe something that doesn't need a new word. Or perhaps means something else. If so can someone explain?
Informally, we say that a point $p$ is a stationary point of a function $f$ if $f(p)$ doesn't change if we move $p$ infinitesimally. Somewhat more formally, the derivative of $f$ with respect to $p$ should be $0$.
This transfers to functionals, e.g. $F[f] = \int_{\mathbb R^n} L(x, f(x), f'(x)) \, dx$. Here the argument is a function $f$ so that is what is to be varied. So how do you vary a function? A more precise definition is that we should have $$\left. \frac{d}{d\lambda} F[f+\lambda\phi] \right|_{\lambda=0} = 0$$ for all $\phi$ in some subspace of the domain (i.e. in the space where $f$ lives).