Action (physics)

See principle of least action.
See also Hamiltonian systems in contact geometry.

What is the action, intuitively?

I see it as the amount of thing that has been done when the system flows. For example, when a system given by a particle in two dimensions is in a state with momentum $p=(1,0)$, a velocity vector (virtual displacement) of $v=(0,1)$ would yield a infinitesimal action

where $\lambda$ is the tautological 1-form. This means that it is not probable for $v$ to be the real direction of movement, since it wouldn't provide so much to the total action

and the principle of least action establish that the total action must be maximized/minimized (stationary).

See variational principle for Hamiltonian mechanics.