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

Motion Equations of Linear Systems

This section provides a quick introduction of motion equations of linear systems, which are first order linear differential equations of the canonical coordinates.

**What Is Linear System?**
A Linear System is a system where motion equations are
first order linear differential equations of the canonical coordinates.
In this case, motion equations can be expressed in the vector form as
shown below:

x' = Ax + b (P.12) where: x is the vector of [q_{1}, q_{2}, ..., p_{1}, p_{2}, ....] x' is the vector of dx/dt A is a matrix of coefficients b is a constant vector

For a Linear System of a single object with 1 degree of freedom, vector x is [q, p] and x' is [q', p']. The motion equations can be expressed as:

|q'| = |a_{11}, a_{12}| . |q| + |b_{1}| |p'| |a_{21}, a_{22}| |p| |b_{2}|

Now let's look at those systems of motions presented earlier:

1. The Free Fall Motion of a single object is a linear system. The motion equations can be expressed as:

m*g = -p' (P.4) p/m = q' (P.5) In vector form: |q'| = |0, 1/m| . |q| + | 0| |p'| |0, 0| |p| |-m*g|

2. An object on a spring moving horizontally is a linear system. The motion equations can be expressed as:

kq = -p' (P.7) p/m = q' (P.8) In vector form: |q'| = | 0, 1/m| . |q| + |0| |p'| |-k, 0| |p| |0|

3. A simple pendulum is not a linear system, because p' is not linearly related to q:

m*g*l*sin(q) = -p' (P.10) p/(m*l*l) = q' (P.11)

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