classical mechanics - What is the physical content of the principle of least action?

Say the world is governed by the Principle of Least Action (or Hamiltonian mechanics) and let's not worry about quantum mechanics too much.

Independently of any Lagrangian or Hamiltonian, does that tell us anything about the world? If yes, what?

To put it differently, is it possible to falsify the Principle of Least Action? What kind of experimental results would do so?

Answer

OP's question seems to be essentially a version of the inverse problem for Lagrangian mechanics, i.e. given a set of EOM$^1$

$$E_i(t)~\approx~ 0,\tag{1}$$ does there exist (or not) an action $S[q]$ such that the EOM (1) are the Euler-Lagrange (EL) equations $$\frac{\delta S}{\delta q^i(t)}~\approx~ 0,\tag{2}$$ possibly after rearrangements? This is in general an open problem. See however Douglas' theorem and the Helmholtz conditions mentioned on the Wikipedia page.

Physically, in the affirmative case, there is a functional Maxwell relation

$$\frac{\delta E_i(t)}{\delta q^j(t^{\prime})}~=~\frac{\delta E_j(t^{\prime})}{\delta q^i(t)}. \tag{3}$$ See also e.g. this related Phys.SE post.

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$^1$ The $\approx$ symbol means an on-shell equality, i.e. equality modulo EOM.