I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange equation" ) and I cannot seem to understand the logic behind it.
So we begin by establishing a functional
$$L = L(x,y(x), y'(x))$$
And we are looking to find the minimal or maximal function that satisfies:
$$ \int_{a}^{b} L(x,y(x), y'(x)) \space dx = H(x)$$
At this point we can consider such an optimal function which we denote as
$$G$$ and therefore note that any solution to the aforementioned will be of the form
$$G(x) + en(x)$$
where e is the magnitude of the error and $n(x)$ is an error function such that $n(a) = n(b) = 0$
At this point what exactly is it that we are trying to do? and more importantly, WHY?