Resources for the free fall periodic solutions
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 into a small window (top left). The program 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).

3) 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 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

Xiaoming Li, Shijun Liao
Collisionless periodic orbits in the free-fall three-body problem
https://doi.org/10.1016/j.newast.2019.01.003

Xiaoming Li, Xiaochen Li, Linghui He & Shijun Liao
Triple collision orbits in the free-fall three-body system without binary collisions
https://doi.org/10.1007/s10569-021-10044-6