In classical mechanics the dynamics on a manifold $M$ are characterised by the minimisation of a functional $$ \min_{q \in C^\infty(\mathbb{R},M)} \int_{\mathbb{R}}L(q(t),\dot{q}(t))dt, $$ where $L:TM\to\mathbb{R}$ is a smooth function, the "Lagrange function". In the case of a particle of mass $m$ constrained to a submanifold $M\subseteq \mathbb{R^n}$ moving in a potential $V: M \to \mathbb{R}$ the Lagrangian is given by $L(x,v):=\frac{1}{2}m\|v\|^2 - V(x),$ so the functional to minimise (the "action") is $$ C^\infty(\mathbb{R},M)\ni q \mapsto\int_{\mathbb{R}}\left( \frac{1}{2}m\|\dot{q}(t)\|^2 - V(q(t)) \right)dt \in \mathbb{R}.$$ This is all standard in physics and many "derivations" of this from variants of Newton's second law on $M$ are given in the physics literature. What I've never seen attempted is an explanation of this: What is the intuition behind formulating the dynamics in terms of a variational principle? Are there any heuristics on the interpretation of the action?

It seems that the action is some sort of "currency": Every path in $M$ comes at a cost and nature somehow chooses that path that costs the least amount. But how to interpret this amount? Why should the currency take the form:

$$\text{"kinetic energy} - \text{potential energy"} ?$$

*EDIT 1:* I'm not looking for a "historical explanation" on how this principle was discovered. I'm more looking for a direct interpretation of the action if this makes sense.

*EDIT 2:* As pointed out by arsmath the dynamics needn't be characterised by a minimum of the action, it can also happen that merely the derivative of the action vanishes and we have a stationary point (or a maximum!).

mechanical activityof the system at time $t$. A ball sitting on top of a hill is intuitively less active that the same ball rolling down the hill, so it makes sense that kinetic energy contributes positively and potential energy contributes negatively to the activity. $\endgroup$ – pregunton Apr 7 at 9:51