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I have been working on the three-body problem lately and came across KAM stability. I read that KAM stability generally means that the solution is stable at different initial conditions (that of course must be close to the initial conditions of the periodic solution).

My question is what does stable here actually means? if I'm given, for instance, the figure-8 solution, how can I determine if it is KAM stable or not by varying the mass ratio and observing the trajectory of all 3 bodies? Can it even be observed graphically? I'm really confused and would like someone to further explain my questions.

Qmechanic
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Belal Bahaa
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1 Answers1

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Normally, KAM stability is about preserving invariant tori as explained in your linked wikipedia article. It usually applies to generic tori due to the various nondegeneracy conditions. Therefore, I don't think it would apply to specific exotic trajectories like the the figure 8 one.

Usually the approach for finding these trajectories are topological. As described in Moore's article "Braids in Classical Mechanics" (Physical Review Letters. 70 (24): 3675–3679), they obtained the trajectories by imposing the topology of the trajectory and minimising the action to get a physical path. The wikipedia article on the three body problem summarises many special solutions. The linked articles would be a good starting point.

Hope this helps.

LPZ
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