Exploratorium
Lab Works in Applied ODEs

The full list of about 200 examples in various fields pre-loaded in this software may be found in the list of Samples (especially numerous in Celestial Mechanics). There, however, it is explained only what those samples demonstrate, yet no didactical material is offered.

On the contrary, this section is intended as a series of lectures on various topics of applied ODEs (and pure mathematics) illustrated with the dynamic simulation of the respective motion. As of this moment (October 2022) this project is only in an initial stage.

You have to download and install the software of the Taylor Center as explained here. This software runs the library of examples of various physical processes as real time animation in 2D or 3D stereo for viewing via a pair of Red/Blue glasses. (3D stereo appears on black background). If you do not have Red/Blue glasses, you may watch it in conventional axonometry (clicking the respective button).

In order to learn and experiment with the simulations and real time animation here, choose and click at the desired title in the list below. This will lead you to downloading a zipped file. Unzip and place it anywhere preserving the structure. It consists of a pdf file with a lecture and explanations what to do and to expect, and several folders with the script files for the software. While reading the pdf, when you decide to experiment with a particular script file mentioned there, run the software and under File/Open Script navigate to the desired script file (or follow the methods of automation mentioned in the Appendix of the pdf files). Then Play it putting the glasses on.

General

Using the Taylor Center to Teach ODEs

Examples of numerous applications

Mechanics

Three Rigid Body motions:

Rolling disk (including the Euler disk), Free body (including the Dzhanibekov effect), and Heavy top

The Guide and didactic manual

The simulations
Video clip of the Dzhanibekov effect  covered in the Guide above

The Spherical Double Pendulum in 3D:

The ODEs of the Spherical Double Pendulum

The simulations

Celestial Mechanics

1) Three body free fall periodic orbits: new remarkable features
The same text in pdf

(the article and the links to the supporting simulations)

Dropping bodies – by Richard Montgomery (pdf)

2) A remarkable mapping into the shape space (w₁, w₂, w₃) and shape sphere (u₁, u₂, u₃) for the 3-body problem
The Guide and didactic manual
The simulations

3) The Continuous Measure of Symmetry as a Dynamic Variable

by Mark Frenkel et al., Symmetry 2023, 15, 2153

Pure Mathematics

Powers which commute or associate as solutions of ODEs

(the article and the script files of the supporting simulations in Appendix 1)

A remarkable periodicity in a real valued extraction of a well-known complex function

(the article and the script files of the supporting simulations in Appendix 2)

Look in Samples for an annotated list of other examples (without didactical explanations).

Look in Syllabi for an annotated list of suggestions and ideas for utilizing these  Samples for writing the respective textbooks.