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. The
loads the ODEs for the 3 body problem with the initial values
corresponding to the selected periodic orbit, compiles, and opens the
Graph window for visual integration. At that, the length of the period
this sample is entered into the termination condition and the program
is set to integration until this termination point. (The period of the
visible also in the Front panel in the Constant section as a comment line
for constant a).
3) In the Graph
window you will see already a result of integration of 10 steps (by
10 steps may not be enough for reaching the terminal point. You
may need to click the
button More several times
changing the number of steps to something bigger, say 100. For some
samples with long periods (over 80) you may need more than 1000 steps.
default the storage is reserved for 1000 points,
therefore when the program asks you to append the storage, do it). You
may notice that graph has not enough room (in default setting). Any
moment you can
adjust the room clicking at the button Adjust.
4) Finally when you obtained the full graph of this periodic
may wish to Play it
dynamically. The default time 5 sec is too short. Enter a longer
interval. Depending on the complexity of the curve, it may be
something like 60-80 seconds. Enjoy the show, and then repeat
everything from step 1 for another sample.
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.