What is Action?

Question

I'm a mathematics student and while I was studying classical mechanics I could find some relationship with Differential Geometry (classical one, with Gauss mapping), but I couldn't understand what is Action properly and the integral definition don't give me any intuition.

Answer 1

Mathematically, the action is defined as being the integral over time of the Lagrangian. So it is a functional over paths that describe what is the motion of the system over a particular path. Also, it has the units of [energy]$\cdot$[time]. Properly, we say that the system must follow the principle of least action, which says that the action must be stationary for a physical path. Since one can show that the equation of motion leads to Newton's formalism if $\mathcal{L}=T-U$, it is accepted that different Lagrangian leads to different formalism, depending on the symmetry one impose on the system. For example, for some region of spacetime on a globally hyperbolic manifold $M$ equipped with a metric, the action for a scalar field would be:

$$ S=\int \mathrm{d}^4x\sqrt{-g}\left(-\frac{1}{2}D_\mu\phi D^\mu\phi-\frac{1}{2}m^2\phi^2\right). $$