Physics Notes - Herong's Tutorial Notes - v3.25, by Herong Yang
Hamilton's Principle - Stationary Action
This section provides an introduction of Action, an integral of Lagrangian function of a given system between two time instances.
What Is Hamilton's Principle? Hamilton's Principle states that the true position function x(t) of a mechanical system is a stationary point of the Action S[x(t)] functional.
Note that Hamilton's Principle is called Stationary Action in some text books.
There are 3 possible scenarios where function x(t) becomes a stationary point for the Action S[x(t)] functional: minimum point, maximum point, and saddle point in a local area.
Some text books also call Hamilton's Principle as Principle of Least Action, with the assumption that the stationary point is always a minimum point.
If x(t) is a stationary point of S[x(t)], we can also say that all small changes of x(t) will result the same value of S[x(t)]. That means S[x(t)] is stationary in the neighborhood of x(t).
Mathematically, a stationary point of Action S[x(t)] functional is a function x(t), where the functional differential of S[x(t)] is zero. This can be expressed as:
d(S[x(t)]) / d(x(t)) = 0
# functional differential of S[x(t)]
or:
d(∫ L(x,x',t)dt) / dx = 0
# Replaced S with its definition
Table of Contents
Introduction of Frame of Reference
Introduction of Special Relativity
Time Dilation in Special Relativity
Length Contraction in Special Relativity
The Relativity of Simultaneity
Minkowski Spacetime and Diagrams
Action - Integral of Lagrangian
Action - Functional of Position Function x(t)
►Hamilton's Principle - Stationary Action
Other Proofs of the Lagrange Equation
Lagrange Equation on Free Fall Motion
Lagrange Equation on Simple Harmonic Motion
Lagrangian in Cartesian Coordinates
Lagrange Equations in Cartesian Coordinates
Introduction of Generalized Coordinates