What Is Phase Space

This section provides a quick introduction to Phase Space, which is the collection of all possible sets of canonical coordinates (q,p) of a system.

What Is Phase Space? Phase Space is the collection of all possible sets of canonical coordinates of a system.

What Are Canonical Coordinates? Canonical Coordinates are generalized position components, qi, extended with generalized momentum components pi.

In other words, a point in Phase Space is a set of canonical coordinates with components of q1, q2, ..., p1, p2, ....

A point in Phase Space is also called the state of the system at a given time instance.

For a single-object system with 1 degree of freedom, the Phase Space is a 2 dimensional space of (q1, p1). In this case, the Phase Space is called Phase Plane.

For a single-object system with 2 degree of freedom, the Phase Space is a 4 dimensional space of (q1, q2, p1, p2).

Note that some text books define Phase Space as the collection of all possible sets of generalized position components qi, with generalized velocity components q'i.

The difference is only a factor of mass, m, in most cases, because of the following relations.

pi = m*q'i

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

 Introduction of Generalized Coordinates

Phase Space and Phase Portrait

What Is Phase Space

 What Is Phase Portrait

 Phase Portrait of Simple Harmonic Motion

 Phase Portrait of Pendulum Motion

 Motion Equations of Linear Systems

 Phase Portraits of 2-D Homogeneous Linear Systems

 References

 Full Version in PDF/ePUB