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 with two degrees of freedom is in a state with momentum $p = (1, 0)$ , a velocity vector (virtual displacement) of $v = (0, 1)$ would yield a infinitesimal action

λ (v) = p d q (v) = 0

where $λ$ 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

S = \int λ,

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

See: tautological 1-form#Physical Intuition The "Action" of a Nudge.

See variational principle for Hamiltonian mechanics.

Economic interpretation: In Hamiltonian mechanics applied to markets, the action $S = \int (p_{a} d q^{a} - H d t)$ represents the accumulated money through an economic trajectory, where $p$ are prices and $q$ are inventories.