I'm not a physicist, and I'm simply studying Lagrangian and Hamiltonian mechanics as examples of calculus of variations and optimal control theory.
So from my perspective, I see the classic principle of least action in physics as simply a special case of the calculus of variations: Newton's law $F=ma$ simply turns out to have has as a consequence that if we define the lagrangian $L$ of a functional to be equal to the kinetic energy minus the potential energy, then minimizing that functional is equivalent to Newton's law.
From my perspective, this is a pure coincidence without any meaning, and therefore there should not be any reason to expect that this particular functional or its Lagrangian $L=K-U$ has any meaningful interpretation.
Yet, physicists have given a name to this functional: "Action". This implies that they give some physical interpretation to it that is meaningful, beyond its technical utility.
So is there a meaningful physical interpretation of the "Action" functional, or of its Lagrangian?
