06 Sept 22 - Notes: The SHO and Phase Portraits#

These is the summary of the PHY 415 class period on 01 Sept 22. If I missed anything, send me an message on Slack and I will add it here. -DC

Today, we worked with the simple harmonic oscillator’s general solution to find particular solutions and remind ourselves how that all works. We used that to talk about 2nd order systems a bit more generally, which is the focus of your first project (a 2nd order ODE of your choosing). We also discussed phase portraits and what they buy us. We will investigate them further with more real systems like the pendulum that can go over the top and the damped oscillator.

Resources#

Here are scans of four sections of four books (that you can look over at my office, or find elsewhere) that are useful for our study of the SHO and will help you with your project ideas and plans.

Sorry for the potato quality, and apparently the orientations, which I corrected but github refuses to keep the same orientation. I will do better next class and try to fix these later.

  • Boas, Mathematical Methods, Sec 8.5

    • This section goes into all the math (in general) for ODEs like the SHO, which are second order (\(\ddot{x}\)), have constant coefficients (e.g., no explicit time dependence of the coefficients), and that are “homogenous” (i.e., have no constant terms or explicit functions of time).

    • If you want to be reminded about all the ODE things, it’s a good read.

  • Crawford, Waves, Secs. 1.1-1.2

    • This is an excellent book in general, but this a good reminder of all the things we did today in book form. His analysis and presenttation are formal, but really jammed packed with information. Crawford doesn’t waste words. This book is out of print, but really worth reading if you are interested in waves. I tried to reproduce some of this in my own notes below.

  • Marion and Thorton, Classical Mechanics, Secs. 3.1-3.2

    • This is a canonical text, that is boring. But it was all the things we did today and some stuff about the energetics.

  • Strogatz, Nonlinear Dynamics, Sec. 5.1

    • This book is great. We are jumping into the middle of it, but the description of phase space analysis is so important and useful. I have tried to reproduce this in my own notes below.

Handwritten Notes#

Below, I have worked up some additional notes that describe the conceptual and procedural aspects of what we did today. Think: derivations and examples for learning from for your project. They are linked below: