Some people see a mystery at the point where these mathematical constructs are applied to our physical universe. Eugene Wigner wrote about “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

There are explanations for many of the points he raised in that paper. Perhaps the one that remains most unresolved is the question of why universal law or behavior isn’t messier, more chaotic - why it should so often correspond so neatly to physical phenomena. Intuitively, this doesn’t seem surprising to me, but Wigner correctly points out that we don’t really know why this is the case.

Answering that gets deeper into philosophy: structural realism, the anthropic principle, and so on. But one possible explanation is an extension of ideas like Noether’s: that the various mathematical constraints collapse the space of possibilities enough to make it likely, if not inevitable, that the universe ends up embodying relatively simple mathematical structures.

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 Classical mechanics has been superseded by Quantum mechanics, with classical mechanics as limiting case; the validity of classical mechanics must be attributed to classical mechanics emerging from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.