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There is a formulation of the classical 3-body Newtonian problem reducing this problem to 6 equations (e.g. http://www.3bproblem.blogspot.com). I would like to know where can I find a discussion or demo of this setting. Any examples of other non-trivial frames for this problem are also welcome, i.e. others than the center of mass frame in 9 equations.

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A general discussion -- including introductions to different formulations -- of the three-body problem can be found in M. Gutzwiller's Chaos in Classical and Quantum Mechanics, Chapters 4 and 5. The references therein may be more useful than the text itself.

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