>>13443614The green graph represents a family of curves that intersects the points 0 and 1. We want to find the curve that has the shortest length between 0 and 1. So, what we do is, instead of comparing the functions directly, we find the arc lengths of the functions (which are constants) and then plot the arc lengths of the functions with respect to a dummy variable. This is where the "variations" part comes from. We find some way to force the functions into constants so that we have some common metric to compare them with. We then introduce a dummy parameter that acts as a new variable, which we then vary along our function. The integral combines all of the results into a single function. If there is an optimal point, some certain value of that dummy variable will result in an integral with a minimum (or maximum) value.
This leads to a second graph, the blue graph. It is the arc lengths of all of the possible curves under consideration. This graph changes direction multiple times, so it stands to reason that it has at least one critical point. The local maximum we can ignore. Through analysis or inspection, we find that the local minimum has a value of sqrt(2) - and it happens when our parameter = 1. x^(1) is just x, which is a straight line.
Lagrangian/Hamiltonian mechanics uses the "Principal of Least Action" to find an optimal function given some constraints. The PLA essentially says "assume whatever phenomenon we're considering will try and find the path of least resistance between two points". This gives us two things: we now have boundary conditions, and we now know that the solution to our mechanical system will be the minimal function of some family of functions. Thus we can use the CoV to find, for example, the path a particle will take down a hill under the influence of gravity since it will follow the "most efficient" path.
