The "three-body" problem and Euler's collinear solution
The "three-body" problem and Euler's collinear solution
Notes
DISCLAIMER
Due to the complexity of the computation it's highly advisable you download the applet on a PC and run it with Geogebra classic 5 for desktop.
It may happen that, upon changing some parameters, some element crashes and/or disappears. Use the applet at your own risk and keep a working copy before making modification.
The applet has not been cleared up and optimized, so following the code and the relations between the elements contained in it could be not simple.
NOTES
From order to chaos.
It all starts like a perfect day. Everything's in order, peaceful, regular and predictable. But then, without any warning sign, the disaster creeps in and everything turns awry. Quarrels arise. Havoc reigns! (damn butterfly!).
In the Euler's solution to the three-body problem the bodies start on a line.
They can have different masses and their motion will be driven only by their mutual attractive gravitational forces, all pointing to the CM (Center of Mass) of the system.
With proper initial conditions Euler (De motu rectilineo trium corporum se mutuo attrahentium - 1767) showed that the three bodies follow three co-focal and co-periodic elliptic orbits in which the common focus is the CM.
Here's an english translation of Euler's article: http://eulerarchive.maa.org/docs/tran...
Euler solution is beautiful and fascinating but, unfortunately, it's also very unstable and the initially ordered configuration soon turns into chaos.