Exploratorium
Lab Works in Applied ODEs
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An annotated list of over 200 simulations in applied ODEs of various fields comes together with the Taylor Center software in a file Samples . (There simulations in Celestial Mechanics are especially numerous). However, there is no didactical material in that file.
The file Syllabi offers an annotated list of suggestions and ideas how to utilize those Samples for your lectures and include them into the respective courses and textbooks.
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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
The Three Classic Rigid Body motions:
the Rolling disk (including the Euler disk),
the Free body (including the Dzhanibekov effect), and the Heavy top
A Course and Didactic Manual
The simulations
Video clip of the Dzhanibekov
effect covered in the Didactic Manual
above
The Spherical Double Pendulum in 3D
A comprehensive course and new cases of uniform motion
with real-time simulations
Celestial Mechanics
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
The Continuous Measure of Symmetry as a Dynamic Variable
By Mark Frenkel et al., Symmetry 2023, 15, 2153
The new samples of the Sitnikov formation motion
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Newest discoveries of remarkable solutions in the three-body problem
Planar setting (in 2D)
Closed periodic orbits Resources on discovery of periodic orbits Run DEMO Menu: 3 bodies / Periodic (closed)
Relatively periodic orbits Resources on discovery of periodic orbits Run DEMO Menu: 3 bodies / Relatively periodic
Free fall periodic orbits (not closed, beginning and ending with points of rest) Three body free
fall periodic orbits: new remarkable features (the article and the links to the supporting simulations) Dropping bodies – by Richard Montgomery (pdf) Run DEMO Menu: 3 bodies/Periodic (free fall) for 30 orbits with equal masses, or… Download and unzip the data, run the menu Set n-body problem / Choose from a list, navigate to and open the file FreeFallHeader.txt in the set of unzipped files, and choose from 234 orbits.
Choreography (all bodies move alone one periodic orbit) Resources on the concept and discovery of 2D Choreographies Run DEMO Menu: 3 bodies / Choreography
Spatial setting (in 3D)
Resources and discovery of 3D periodic orbits and choreographies. Download and unzip the data, run the menu Set n-body problem / Choose from a list, navigate to and open the file Periodic3DHeader.txt in the set of unzipped files, and choose from 31 orbits.
Discovery of 3D periodical orbits for Four-body problem Run DEMO Menu: 4 bodies / Cubic
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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.