Stationary Action

What to observe

1. The two ends are pinned and everything between them is yours to choose. Bend the path and the action gets bigger; bend it the other way and it gets bigger again. The real path sits at the bottom of that valley, and that is the whole principle: not the smallest path, the one where small changes stop mattering.
2. Look at the curve of S against how hard you bend. At your path the slope is some number; at the real path the slope is zero. This is what "stationary" means, and it is why the same idea keeps working when the real path is a maximum or a saddle instead of a minimum.
3. The path is painted with its own Lagrangian: orange where the kinetic term is winning, blue where the potential is. Bending upward buys more time in the blue, which subtracts less from L, but you have to move faster to get there and back, which adds more orange. The real path is the exact trade between the two, and it is a parabola only because that trade balances where it does.
4. In the Lagrangian panel, L is the difference of the two thin curves and the shaded area under it is the action. Bend the path and watch the shape of that area change: everything the principle talks about is the size of that patch, nothing else.
5. Keep an eye on the dashed line for KE + PE. On the real path it is dead flat, because energy is conserved. On any path you bend it visibly wobbles, which is the sim telling you an invisible hand would have to push the bead to make it go that way. L is not energy, and the flat one is the sum while the interesting one is the difference.
6. Switch to Nothing and the trade disappears: with no potential the only way to spend less is to not hurry, so the answer is a straight line at constant speed. Newton's first law falls out of a principle about a whole path.
7. Turn on the beads and race them. Both leave and arrive together because both are forced to, but only one is moving the way a real object would. The dots along each path mark equal slices of time, so where they spread out the bead is fast: that spacing is the kinetic term made visible.
8. Press "Let it relax" and watch the path slide downhill in action. Every step it takes is ma = −dV/dx being enforced a little more, because the gradient of the action is precisely that error. Minimising the action and obeying Newton are two descriptions of one thing.
9. Try the spring and then stretch the trip time. The valley in the S curve gets shallower, flattens, and eventually turns into a hill: past a certain trip time the true path stops being the smallest action and becomes a saddle. Nature never promised the smallest, only that the first-order change vanishes, which is why the principle survives this.