I have seen that not every function $q(t)$ has a corresponding $\mathcal{L}$ such that $\frac{\partial \mathcal{L}}{\partial q}=\frac{d}{dt}\frac{\partial \mathcal{L}}{\partial \dot{q}}$, and I am assuming the same applies for fields. I am wondering, then, what properties a given function or field is guaranteed to have if it does have a corresponding Lagrangian. Given how generally the principle of least action is applied, I am assuming it enforces constraints that are somewhat physically intuitive. I have seen online that it requires "stability under perturbations"--I didn't find much of an elaboration but by the sound of it it requires that $\mathcal{L}+\delta\mathcal{L}$ has minimal action with the path $q+\delta q$ (or field $\phi+\delta \phi$), so perturbations are of similar order. I am wondering if anything else that is intuitive/motivating for the principle of least action can be said.
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