for the free fall periodic
of plane 3 body problem
The 3- and n-body
problem has been a challenge for analysis. In most cases of planar 3
body problem with equal masses and resting center of mass the
trajectories look chaotic, or they end up in two bodies coupling and
moving to infinity, while the third body moving also to infinity in the
Discoveries of periodic solutions of such 3-body problems was a
challenge solved in the modern time with the help of computers. Xiaoming
LI and Shijun LIAO from the Shanghai Jiaotong University,
China, discovered hundreds of periodic orbits which happened to be not
closed curves, but finite curved segments at whose extremes the bodies
have zero velocity (i.e. they are in a free fall).
conditions for 30 such free fall periodic solutions were entered into
the Demo of the
Taylor Center each of which can be
loaded and played here. Here is how.
1) Go to Demo/3 bodies/Periodic (free fall)/A list
integrate (not a script)
2) Enter a number of the desired sample between 1 and 30 into a small window (top left).
loads the ODEs for the 3 body problem with the initial values
corresponding to the selected periodic orbit, compiles, and (blindly) integrates the problem until
reaching the termination point – the period of this simulation entered
from its file. (The period of
the orbit is visible also in the Front panel in the Constant section
as a comment line for constant a).
When the integration reaches the termination point (the period), the
program displays the message. As you click OK, the program opens the
Graph window displaying the entire trajectory. You may wish to Play it
dynamically (by default the duration is 25 s). Depending on the
complexity of the curve, it may be
something like 60-80 seconds. Enjoy the show, and then repeat
everything from step 2 for another sample.
4) If you like the
curves that you have obtained and played, you may wish to save this
session as the script so that next time they integrate ready for
playing automatically. In the File menu, Save Script into
a location of your choice (available only in the licensed version).
Next time you will be able to open this script and play it.
Xiaoming LI and Shijun LIAO, Movies of the Collisionless Periodic
Orbits in the Free-fall Three-body Problem in Real Space or on Shape
Xiaoming LI and Shijun LIAO, Collisionless periodic orbits in the
free-fall three-body problem.
Xiaoming Li, Shijun Liao
Collisionless periodic orbits in the free-fall three-body problem
Xiaoming Li, Xiaochen Li, Linghui He & Shijun Liao
Triple collision orbits in the
free-fall three-body system without binary collisions