Resources
for the free fall periodic
solutions

of plane 3 body problem

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 opposite direction.

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).

The initial 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
to
integrate (not a script)

2) Enter a number of the desired sample between 1 and 30. The program 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 of this sample is entered into the termination condition and the program is set to integration until this termination point. (The period of the orbit is 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 default). 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. (By 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 orbit, you 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.

2) Enter a number of the desired sample between 1 and 30. The program 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 of this sample is entered into the termination condition and the program is set to integration until this termination point. (The period of the orbit is 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 default). 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. (By 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 orbit, you 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 Sphere

http://numericaltank.sjtu.edu.cn/free-fall-3b/free-fall-3b-movies.htm

Xiaoming LI and Shijun LIAO, Collisionless periodic orbits in the free-fall three-body problem.

https://arxiv.org/pdf/1805.07980.pdf