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
Famous quotes containing the words general and/or statement:
“Women born at the turn of the century have been conditioned not to speak openly of their wedding nights. Of other nights in bed with other men they speak not at all. Today a woman having bedded with a great general feels free to tell us that in bed the general could not present arms. Women of my generation would have spared the great general the revelation of this failure.”
—Jessamyn West (1907–1984)
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)