**Physics Notes - Herong's Tutorial Notes** - v3.24, by Herong Yang

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, q_{i},
extended with generalized momentum components p_{i}.

In other words, a point in Phase Space is a set of
canonical coordinates with components of q_{1}, q_{2}, ...,
p_{1}, p_{2}, ....

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 (q_{1}, p_{1}).
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 (q_{1}, q_{2},
p_{1}, p_{2}).

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

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

p_{i}= m*q'_{i}

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

Introduction of Generalized Coordinates

►Phase Space and Phase Portrait

Phase Portrait of Simple Harmonic Motion

Phase Portrait of Pendulum Motion

Motion Equations of Linear Systems