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

What Is Phase Portrait

This section provides an introduction to Phase Portrait, which is the trajectory curve of a system in the the Phase Space for a given period of time.

**What Is Phase Portrait?**
Phase Portrait is the trajectory curve of
a system in the the Phase Space for a given period of time.

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 Portrait becomes a 2 dimensional curve.

Phase Portrait of Free Fall Motion

Let's take a look at Phase Portrait of the Free Fall Motion of a single object with mass m. In this case, the generalized position has only 1 component x, representing the height of the object. So we can express Canonical Coordinates (q,p) of the system as below:

q = (x) p = (m*x') # x is the height of the object # m is the mass of the object # x' is the velocity of the object

The Hamilton Function can be expressed as:

H = T + V or: H = p*p/(2m) + m*g*q (P.1) # g is the standard gravity (9.80665)

If we apply Hamilton Equations, we have:

∂H/∂q = -p' (P.2) ∂H/∂p = q' (P.3) or: ∂(p*p/(2m) + m*g*q)/∂q = -p' ∂(p*p/(2m) + m*g*q)/∂p = q' or: m*g = -p' (P.4) p/m = q' (P.5)

Equations P.4 and P.5 are the equations of the free fall motion, which has the following solution:

q(t) = -g*t*t/2 + p_{0}*t + q_{0}p(t) = -m*g*t + p_{0}# q_{0}is the initial position # p_{0}is the initial momentum

The Phase Portrait of this system is a parabolic curve in the Phase Plane of (q,p). For example, if the system has a initial condition of (q,p) = (0,1), its Phase Portrait will look like this:

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