Why do we need to imagine a varied path in the calculus of variations?

[NihalRi]

I'm very new to this. So in the context of finding the shortest path the idea is that you imagine another path that starts and ends at the same point. The shortest path is a minima so you differentiate and find for what values the differential is zero.
I don't understand why we need to imagine this varied path, why not differentiate the original path?

 

[ShayanJ]

Actually we're differentiating the action w.r.t. to paths. Its pretty much like differentiating a function of one variable. The definition is $\displaystyle f'(x)=\lim_{\delta\to 0}\frac{f(x+\delta)-f(x)}{\delta}$. The analogue of your question about the derivative of a function of one variable is that why are we considering $f(x+\delta)$? Why not only $f(x)$ appears in the definition?
The answer in both cases is that we need to find out in what point the derivative is equal to zero to lowest order, which means the function doesn't change when we move only a little bit. You just need to recognize that an action integral is a function of paths between two given endpoints so when we want to calculate its derivative, we have to consider two nearby paths between those two endpoints(like $x+\delta$ and $x$ above) and calculate the output of the action integral for both and demand that the difference vanishes to lowest order. This way we find out at what path this happens and that'll be the correct path. This is pretty much like setting the first derivative of a function of one variable to zero to find in what $x$s the function has extrema.

 

[Ssnow]

The IDEA is the following: you start with a family of continuous paths from the initial point $A$ to the final point $B$, you can parametrize this family by a parameter $t$ so for each $t$ you have a path between $A$ and $B$, and the corresponding length of the path from $A$ and $B$. So you have a function $\mathcal{L}(t)$ that for each path $t$ give you the length. You minimize the quantity $\mathcal{L}(t)$ studying the derivative and you will find the minimum length, so the minimum path ...