One section of that will be replaced in a day or two: the last part of section 2. I completed a new diagram, that diagram will allow me to cut a lot of text. I believe the change will be a significant improvement.
Repeating the links: Page dedicated to the case of a potential proportional to the cube of the displacement: http://cleonis.nl/physics/phys256/stationary_action.php
From F=ma to Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...
There are other demonstrations available that go from the newtonian formulation to Hamilton's stationary action. I believe the one in my resource is the most direct demonstration. (As in: a more direct path doesn't exist, I believe.)
(If you are interested, I can give links to the other demonstrations that I know about.)