Playing with the DEMO

This is a limited version of the Taylor Center – a sophisticated integrator for ordinary differential equations with powerful graphical features, in particular 3D stereo real time animation. It is loaded with numerous examples of fascinating classical problems, allowing to "play" them in real time motion, to experiment with a 3D cursor, and to study effects of specific parameters.

To install the program, just unzip it into an empty folder of your choice. Then run TCenter.exe and follow the steps guiding you through several examples producing 3D trajectories.

If the top edge of windows on your desktop looks like this image (the underlying icons are partially visible through the edge), this is a new MS Windows visual effect which slows down the visual output  in real time.Since Vista and Windows, a new visual feature Windows Aero (a backronym for Authentic, Energetic, Reflective, and Open) employs an effect of translucency which dramatically slows the process of video output: approximately 10 times.

Therefore, this program switches Off the Aero feature when the program starts, and the operating system switches Aero back when the program ends.

In the start menu select Demo/Three Bodies/Disturbed/3D: it opens the initial value problem script and compiles it. After clicking OK to the message "Compilation successful", some knotty Red and Blue curves will appear. Now put on your anaglyphic glasses (over those you usually use, if any) and get ready for fun. (It's recommended to maximize the Graph window).

What you hopefully perceive looks like a "fishing line" hanging in thin air between the monitor and your face. These are trajectories of three bodies moving under gravitational pull. More specifically, this is the so called disturbed Lagrange case. (In the Lagrange case proper, three equal masses are placed at vertices of an equilateral triangle with initial velocities comprising an equilateral triangle co-planar to the first one – Demo/Three Bodies/ Symmetrical). This "fishing line" is a result of a small disturbances applied perpendicularly to the initial plane (the plane of your screen).

The program is capable of producing something more than "still life". Click the Play button. This initiates real time animated 3D stereo motion of the bullets representing the three bodies with all the accelerations, decelerations, and couplings.

When they come to rest, you may try exploring the elements of the trajectories with a "tactile" 3D cursor. Move it into the scene, where it will transform into a small cross. The mouse always moves the stereo cursor in a plane parallel to the screen. In order to control its depth, use the mouse wheel. Another method of changing the depth is to move the mouse keeping depressed either Ctrl key (to bring the cursor closer to your eyes), or Shift key (to move it away from you). Current 3D coordinates of the cursor always appear at the top window panel.

Now, applying the 3D control, try to touch one of the trajectories in space with the 3D cursor. If the speakers are ON, you will hear a clicking sound when the touch occurs: this is the so called "tactile" audio feedback, helping to explore points of interest in the curves.

You can rotate the curves in the space with the Turn controls. With the given specific sizes of the parallelepiped, you may notice that the front side (controlled by Max Z value) keeps the curves inside the parallelepiped "flattening" them. (Therefore increase Max Z).

Already familiarized with the 3D stereo features of the package, you may try several other problems. Click Main Panel in the menu to re-visualize the main form, and go to Demo/Four Bodies. The two pairs of bodies with equal masses are all initially placed in a horizontal plane, parallel to your desk (perpendicular to the screen). The horizontal components of the velocities provide near circular motion for each coupled pair, while the small vertical components push the two pairs into a large circular motion around the center of the masses (see the initial values in the Main window). At the beginning the trajectories spin into a braid looking like a torus (and like the tiny braided rings of Saturn shot by the Voyager probe), but the braid actually does not outline a torus: you can notice that both coupled pairs preserve their initially horizontal plane.

Another fascinating example of 3D motion is under Demo/Möbius. You can watch 4 bullets lined up in a straight line whose motion outline a Möbius surface winded 1.5. To get a more common one (winded 0.5), change value of n=0.5 (in Constants), Compile, click button Previous (in Graph setting page), click Clear in Graph window, and finally click the More button.   

You can explore several more 3D stereo examples opening them as scripts. Click the Main Panel and go to File/Open script menu item. Here are files producing 3D stereo images:

PendulumApple.scr,  PendulumFlower.scr (spherical pendulum)  

KnotChain3D.scr, TrefoilKnot3D.scr


Beside 3D stereo samples, there are also instructive examples in 2D, such as the recently discovered eight-shaped solution of the three body problem called "Choreography" (Demo/Three Bodies/Choreography). Under File/Open script there are also two more classical examples in celestial mechanics: the Euler case with the bodies of equal masses (3EqBodEuler.scr) and the case when one mass is near zero (3NonEqBodEuler.scr). There are also scripts for single and double pendulums, and the Four body Lagrange case as well.

To obtain the full version of the Taylor center, contact Alexander Gofen, 333 Fell St. #218, San Francisco, CA 94102. Phone (415) 863 5125