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 ... qN) 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).
Read more about this topic: Principle Of Least Action
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