Principle of Least Action - General Statement

General Statement

The starting point is the action, denoted (calligraphic S), of a physical system. It is defined as the integral of the Lagrangian L between two instants of time t₁ and t₂ - technically a functional of the N generalized coordinates q = (q₁, q₂ ... q_N) which define the configuration of the system:

where the dot denotes the time derivative, and t is time.

Mathematically the principle is

where δ (Greek lower case delta) means a small change. In words this reads:

The path taken by the system between times t₁ and t₂ is the one for which the action is stationary (no change) to first order.

In applications the statement and definition of action are taken together:

The action and Lagrangian both contain the dynamics of the system for all times. The term "path" simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. the curve q(t), parameterized by time (see also parametric equation for this concept).