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Even though the historical order is the other way around, it is helpful to start from wave/quantum mechanics and arrive at classical mechanics in the limit that the wave length of the particle goes to zero. Mathematically, that limit is the stationary phase approximation, meaning that classical trajectories are perpendicular to surfaces of constant phase. The phase is the action = integral of Lagrangian $L$, as first realized by Dirac, and as follows immediately by calculating the phase $\phi$ accumulated in a time $T$, $$\phi=\int_0^T(p\dot{q}-H)\,dt=\int_0^T L(q,\dot{q})\,dt.$$ From this equation you see that the Lagrangian is "kinetic minus potential energy" because it is the difference of $p\dot{q}$ = twice the kinetic energy and the sum $H$ of kinetic and potential energy. In this way stationary phase amounts to stationarity of the action.