I have been trying to understand functionals in the sense of variational calculus and there was an example of finding the shortest path between two points.
The arc length integral which is easy to derive measures the distance between those stationary points let us say x1 and x2. The functional is a function where the input is another function.
The solution to this functional is the Euler Lagrange equation which I assume is the differential equation that solves for the arc length integrals that are stationary . ( a min or a max I assume )
So how do we know the arc length integral is the functional that the Euler Lagrange equation is solving or just another instance of one particular length ?
P.S. Maybe I am not understanding the definition of a functional ….it seems to me it's like a composite function but the second function just happens to be an integral that you feed the first function into so they found a new name, functional. Where as in composite function no integral is involved but it's still the same business, a function gets fed into anther.
Best Answer
To summarize the comments above: The arc-length integral defines a length. If you consider it as a functional then plugging in any function and computing the integral will give you the arc-length of this function.
If you only care about the arc-length of a given function then you don't need to consider the concept of a functional at all, you simply compute the arc-length integral for your function (and the Euler-Lagrange equations are not relevant).
The concept of a functional is useful when we consider problems like "what is the function that has the shortest arc-length between two points", i.e. when the solution we are after is a function. Then we would start by putting up a functional that describes the quantity we want to extremize (the arc-length), derive the Euler-Lagrange equations for this functional ($f''(x) = 0$) and solve it to get the desired function (a straight line connecting the two points).