Qns on euler-lagrangian equation

[almo]

I find it hard to undestand the various notation used for the equation.
Am i wrong to understand the equation as finding the maxima or the minima of an function?
However, the terms like functional and small real parameter confuses me.
I read up on what's a functional and can't really understand, so far my understanding of its, is that its a function where by instead of x, a varible, it consist of vectors like velocity and etc. Thus, am i wrong to say equation of KE is actually a functional?
On the part of small real parameter ε.. i just have no idea. All i can infer is that is a change in the vector. But where is there this need to implictly express such a term?
Is euler-lagrangian eq considered as tough for an undergrad?
i am seriously struggling with it...

 

[Vargo]

Do you know what a vector space is? A functional is a map whose domain is a subset of a vector space and which takes scalar values.

In the context of your question a typical vector space would be the set of differentiable functions on the interval [0,1].
$V =\{ y(x)| y\, \text{is differentiable in a neighborhood of the interval}\, [0,1]\}$

An example of a functional would be a map $\mathcal{F}(y)$ with domain
$\{y\in V| y(0)=1,\, y(1)=5\}$ and which is defined by a formula such as
$\mathcal{F}(y) = \int_a^b \sqrt{1+(y')^2}\, dx$

In plainer language, in the context of calculus of variations, functionals take ordinary functions as inputs and return numbers as outputs.

A good basic reference would be Gelfand "Calculus of Variations".