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

Phase Space and Phase Portrait

This chapter provides an introduction of phase space and phase portrait. Topics include introduction of canonical coordinates of a system; trajectory curves of a system for a given period of time; phase portraits of free fall, simple harmonic, and pendulum motions; motion equations of linear systems; 2-D homogeneous linear systems.

Phase Portrait of Simple Harmonic Motion

Phase Portrait of Pendulum Motion

Takeaways:

- Phase Space is the collection of all possible sets of canonical coordinates of a system.
- Canonical Coordinates are generalized position components, q
_{i}, extended with generalized momentum components p_{i}. - Phase Portrait is the trajectory curve of a system in the the Phase Space for a given period of time.
- Motion equations in a linear system are first order linear differential equations of the canonical coordinates.
- 2-D homogeneous linear systems can be classified based on characteristic polynomials of their coefficient matrixes.

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