Restricted Three-Body Problem in a Plane

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This Demonstration shows two examples of the restricted planar three-body problem. One is like the Earth-Sun-Moon system, in which two bodies have small mass relative to the third. The second is like the Sun-Jupiter-asteroid system, in which one body has a small mass relative to the other two. In general, such systems will exhibit chaotic behavior.


Several types of trajectories and the approach to the chaotic regime are shown.


Contributed by: Enrique Zeleny (October 2012)
Open content licensed under CC BY-NC-SA



The equations of motion are



where are the reduced masses, while and are the coordinates in the rotating system. The small body is subjected to both centrifugal and Coriolis forces.

In some cases, the third mass appears to orbit around points other than the large masses. These are called "Lagrange points"; there are five such equilibrium states in the co-rotating system.

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