Actions in economics Mirowski 1989 argues economics and physics have frequently informed each other's theoretical development, and neoclassical economics has the same Lagrangian and Hamiltonian formalism frequently seen in physics. I don't know how mainstream this approach has ever been, but his account doesn't make explicit what would be an action in economics. For example, I can imagine a firm's profit maximization, or a portfolio's ROI variance minimization, being formalized as an extremal action principle, but only if such quantities are expressible in the form $\int_a^bL(t,\,q,\,\dot{q})dt$ or a suitable generalization. Are there any specific examples like that?
Select examples from a book suggestion due to @markleeds, Optimal Control Theory with applications in Economics:

Polluting growth has action $\int(bz-\dot{z}+\ddot{z})dt$.
Optimal extraction of a natural resource has action $\int(C(-\dot{x},\,t)+q\dot{x})e^{-st}dt$.
Consumption vs rapid attainment of production capacity has action $\int[\dot{x}-x+q-a]dt$.
Resource extraction in an open economy has action $\int-U(\bar{c}-e^{rt}\dot{x}+ay-\dot{y})dt$.
Consumption versus investment can give rise to many actions, including:

$\int(\dot{x}-1)U(x)dt$
$\int e^{-\rho t}(\dot{K}+\delta K-f)U(K)dt$
$\int\frac{dt}{K-\dot{K}}$
$\int e^{\dot{K}-aK}dt$
$\int(\dot{K}-cK)^ae^{-bt}dt$