[Cleonis]

As to understanding Hamilton's stationary action deeply: that is accessible.

I have created a resource with interactive diagrams. Move sliders to sweep out variation of a trial trajectory. The diagram shows the response.

https://cleonis.nl/physics/phys256/energy_position_equation....

About the form of the resource:

In physics textbooks the usual presentation is to posit Hamilton's stationary action, followed by demonstration that F=ma can be recovered from it.

Now: we have that in physics you can often run derivations in both directions.

Example: the connection between the Lagrangian formulation of mechanics and the Hamiltonian formulation. The interconversion is by way of Legendre transformation. Legendre transformation is it's own inverse; applying Legendre transformation twice recovers the original function.

Well, the relation between F=ma and Hamilton's stationary action is a bi-directional relation too: it is possible to go _from_ F=ma _to_ Hamilton's stationary action.

The process has two stages:

- Derivation of the work-energy theorem from F=ma

- Demonstration that in circumstances such that the work-energy theorem holds good Hamilton's stationary action holds good also.

Knowing how to go from F=ma to Hamilton's stationary action goes a long way towards lifting the sense of mystery.

General remark: Of course, in physics there are many occurrences of hierarchical relation. Classical mechanics has been superseded by Quantum mechanics, with classical mechanics as limiting case; the validity of classical mechanics must be attributed to classical mechanics emerging from quantum mechanics in the macroscopic limit.

But in the case of the relations between F=ma, the work-energy theorem, and Hamilton's stationary action: the bi-directionality informs us that the relations are not hierarchical; those concepts are on equal par.