The comprehensibility of the world is an illusion. Humans don't explain, we describe, and we give ourselves credit for undoing the confusion of some superficial effects that turned out to have a simple description. The mystery and awe of what the universe is doing and how remains poignant.
I have created an educational resource in which I address the question of how it comes about that F=ma can be recovered from Hamilton's stationary action. This resource gives a two-pronged approach: the concepts are illustrated with interactive diagrams, and parallel to that a full presentation of the mathematics.
I start with a discussion of the nature of Calculus of Variations. I use the problem of a soap film stretching between two parallel concentric rings as motivating example. This leads to a derivation of the Euler-Lagrange equation.
Then I move to the Catenary problem. Interestingly, with the catenary problem both approaches are possible; you can solve for the catenary with differential calculus (as Leibniz did) or you can apply calculus of variations. What that means is that the catenary problem can serve as a Rosetta stone, offering a bridge between differential calculus and calculus of variations.
http://cleonis.nl/physics/phys256/calculus_variations.php
The discussion specifically for Hamilton's stationary action is in an article of its own:
http://cleonis.nl/physics/phys256/energy_position_equation.p...
That's the whole point of GR - coordinate systems aren't physically meaningful.