\(S = \int Ldt\), where \(L\) is the Lagrange function and \(t\) is time.
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An action is usually an integral over time. But for action pertaining to fields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system. If the action is represented as an integral over time, taken a the path of the system between the initial time and the final time of the development of the system.
The evolution of a physical system between two states is determined by requiring the action be minimized or, more generally, be stationary for small perturbations about the true evolution. This requirement leads to differential equations that describe the true evolution. Conversely, an action principle is a method for reformulating differential equations of motion for a physical system as an equivalent integral equation. Although several variants have been defined (see below), the most commonly used action principle is Hamilton's principle.