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://fr.wikipedia.org/wiki/M%C3%A9trique_de_Schwarzschild, 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 $c$ $$A=\int mc^2dt -\int \Big(\frac{1}{2}m(\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2})-V\Big)dt+\mathcal{O}(\frac{1}{c^2})$$

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.

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 Schwartchild metric https://fr.wikipedia.org/wiki/M%C3%A9trique_de_Schwarzschild, 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 $c$ $$A=\int mc^2dt -\int \Big(\frac{1}{2}m(\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2})-V\Big)dt+\mathcal{O}(\frac{1}{c^2})$$

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://fr.wikipedia.org/wiki/M%C3%A9trique_de_Schwarzschild, 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 $c$ $$A=\int mc^2dt -\int \Big(\frac{1}{2}m(\frac{dx^2}{dt^2}+\frac{dy^2}{dt^2}+\frac{dz^2}{dt^2})-V\Big)dt+\mathcal{O}(\frac{1}{c^2})$$