If anyone has a reference/book/paper that allows you to learn this concept more intuitively, I'd be grateful.
It is possible to go in all forward steps from F=ma to Hamilton's stationary action; that is what I present.
The path from F=ma to Hamilton's stationary action consists of two stages: (1) Derivation of the work-energy theorem from F=ma (2) Demonstration: when the conditions are such that the work-energy theorem holds good then Hamilton's stationary action will hold good also.
I recommend that you first absorb the presentation of the subset of Calculus of Variations that is applied in physics: http://cleonis.nl/physics/phys256/calculus_variations.php
Discussion of Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...
These presentations are illustrated with interactive diagrams. Each diagram has one or more sliders for manipulation of the contents of the diagram. That way a single diagram can offer a range of cases/possibilities.
About my approach: I think of Hamilton's stationary action as an engine with moving parts. To show how an engine works: construct a model out of translucent plastic, so that the student can see all the way inside, and see how all of the moving parts interconnect. My presentation is in that spirit.