Hamilton's Principle - Stationary Action

∟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

About This Book

Introduction of Space

Introduction of Frame of Reference

Introduction of Time

Introduction of Speed

Newton's Laws of Motion

Introduction of Special Relativity

Time Dilation in Special Relativity

Length Contraction in Special Relativity

The Relativity of Simultaneity

Introduction of Spacetime

Minkowski Spacetime and Diagrams

Introduction of Hamiltonian

►Introduction of Lagrangian

What Is Lagrangian

Action - Integral of Lagrangian

Action - Functional of Position Function x(t)

►Hamilton's Principle - Stationary Action

What Is Lagrange Equation

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

Phase Space and Phase Portrait

References

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