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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?
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.
Physically we have the Newton equations of the second order and two initial conditions. This helps us predict the future positions and velocities.
Unfortunately, the least action principle uses initial and final positions as if they were "known" physically. Thus the physical meaning is quite small and it is actually reduced to the statement that we deal with second order differential equations. No unique evolution is predicted by the equations themselves since they admit two-parametric liberty: the initial position and velocity. The "future variables" (i.e., taken at $t+dt$) are actually determined by the present: $p(t+dt)=p(t) + F(t)dt$, $x(t+dt)=x(t)+v(t)dt$. By the way, these relationships encode the so called "causality".
Falsifying happens when we understand and see the approximative character of these laws. There are some inequalities that make our equations "work". Outside these inequalities the equations do not work. For example, an approximation of point-like particles for finite size bodies, etc., etc.
It would be big news the principle of least action was experimentally falsified. This, though, is unlikely to happen.
Rather we should expect something similar to the principle of least action to always hold.
Recall, that classical mechanics is deterministic and hence specifying that initial conditions we find a determined trajectory through which the system moves.
This can be alternatively phrased as saying there is a space of possible trajectories that a system can move in and specifying the initial condition means we pick out a specific trajectory.
Now, calculus allows us to pick out specific points on a curve where the gradient is at a minimum, these are the stationary points.
Thus we ought to expect that we can find a functional on the space of trajectories whose stationary point is the desired trajectory. This functional is the Lagrangian.
The physical content is that the Lagrangian is built simply from the difference of the kinetic and potential energy of the system and thus takes a very simple form.
It's probably worth pointing out that the principle is of archaic origin: Hero of Alexandria pointed out in 100AD that light reflecting from a mirror follows the path of least distance, this was rephrased by Damianus in 200AD as the path of least time and then much much later, Fermat generalised this in 1657 to any motion of light follows a path of least time.
