One could also mention the Special and General Relativity. The postulate is that the dynamic of a particle is a geodesic. More precisely it minimizes the quantity
$$ A=\int mc\sqrt{ds^2} $$ where $$ ds^2 = (1+\frac{2V}{mc^2})c^2dt^2 - dx^2-dy^2-dz^2 $$
with $V$ is the potential. See for example the Schwarzschild metric ( https://en.wikipedia.org/wiki/Schwarzschild_metric), where the Newton potential appears and we neglected the modifications on the space which are of smaller order for large $c$. Then 
$$ A= \int mc\sqrt{c^2+\frac{2V}{m}-\frac{dx^2}{dt^2}-\frac{dy^2}{dt^2}-\frac{dz^2}{dt^2}}dt $$
and for large $c$ $$A=\int mc^2dt -\int \Big(\frac{1}{2}m(\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2})-V\Big)dt+\mathcal{O}(\frac{1}{c^2})$$
Therefore, the action can be seen as the first order of the path length. 

EDIT: True we start from a minimising principle (geodesic) to obtain another minimizing principle (with the Lagragian), however the first one is somehow more fundamental than the second one.