Action

see also:

• Principle of Least Action
• Principle of Stationary Action
• [Lagrangian Mechanics]
• [Calculus of Variations]
• Conservation Laws
• Noether’s Theorem

The action, $S$, is a functional, which means it takes a function (the system’s path through its configuration space) as input and returns a scalar. It is defined as the integral over time of the Lagrangian, $L$, of the system:

$$S[q]={\int }_{t_1}^{t_2}L(q(t),\dot{q}(t),t)dt$$

Here, $q(t)$ represents the generalized coordinates of the system (which can be positions, angles, or field values), $\dot{q}(t)$ are their time derivatives, and $L$ is the difference between the kinetic and potential energies for classical systems.