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Anyway; my main question has to do with the process of getting the answer you're looking for. Is every problem in calculus of variations possible to be set as a solvable differential equation? And by solvable I don't mean keep guessing the answer until you eventually get it and then prove it was the right one using some method. Which to me looks like a general trend in calculus of variations.

If you can set at least most problems of calculus of variations as differential equations, how do you go about doing it? Should I look deeper into functional analysis, banach spaces, metric spaces and so on?

Could anyone show me an "algorithmic" way to find, for instance, the solution to the problem of finding the shortest path between two point in the xy plane? Without assuming you already know the answer and is just trying to prove it's the right one?