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"Gravity is a harsh mistress." -- The Tick.
When three objects get involved, the equations are much more complicated. Many three-body problems in gravity are still unsolved. Several interesting three body problems have just recently been discovered.
One of them is a figure-8 orbit. You can see details of it at the homepage of Professor Richard Montgomery. Here's the technical details:
Three equal masses chase each other around a figure eight shaped curve in the inertial plane. The three colors (if you can see them) are meant to indicate the three masses. Each mass traversing the corresponding colored arc in one-third of the total period T of the orbit. They start in an Euler configuration with red at the midpoint of blue and green. Call this `R' to indicate the midpoint of red. In time T/3 indicated, B (blue) is the new midpoint, for an Euler configuration lying on the same line. In time T/6 green makes it to the origin resulting in Euler configuration G. In all we get the sequence RGBRGB or 123123 which encodes this orbit. An important fact for our proof is that at time T/12 the configuration is isosceles.In the process of doing more research, I learned that Earth has a second moon/companion. It's called Cruithne.
initial conditions: (x1,y1) = (-0.97000436, 0.24308753), (x2,y2) = (-x1, -y1), (x3,y3) = (0,0)
initial velocites (vx1,vy1) = (vx2, vy2) = -(vx3, vy3)/2; where (vx3,vy3) = (0.93240737, 0.86473146)
masses: all 1. gravitational constant: 1.
(courtesy Carles Simo)
The research: I was websurfing, and found http://www.fourmilab.com/, the website of John Walker. He founded Autodesk. All the diagrams on this site were made with Autocad, so I can be quite grateful. John has a lot of interesting material on physics, astronomy, and business. I was there mainly for his Sky Screen Saver. Also fascinating are the Oh-My-God Particle and Orbits in Strongly Curved Spacetime.
From there, I found The Nine Planets by Bill Arnett. The whole site is fascinating. I didn't think I could learn much from the Earth page, but I was wrong. It contains a link (http://aries.phys.yorku.ca/~wiegert) to research on Asteroid 3753 (1986 TO).
Bob Jenkins had already written a fantastic article on Cruithne, with interactive applets! Consider this link mandatory.
http://www.astro.queensu.ca/~musgrave/cforce/ explores what happens when the laws of gravity are changed.