Action (physics) - Mathematical Definition

Mathematical Definition

Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action.

Several different definitions of 'the action' are in common use in physics:

• The 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.

• 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.

• An earlier, less informative action principle is Maupertuis' principle, which is sometimes called by its (less correct) historical name, the principle of least action.

If the action is represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system,

where the integrand L is called the Lagrangian. For the action integral to be well defined the trajectory has to be bounded in time and space.

Action has the dimensions of •, and its SI unit is joule•second. This is the same units as angular momentum.