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Resolution of the Density Conjecture for Newton's N-body problem

Suggested by Richard Montgomery, UCSC

The motion of the point masses in a gravitational N-body system is "bounded" if all the inter-body distances remain less than some fixed constant for all time. For example, periodic solutions, such as these engaging trajectories, are bounded. A solution is unbounded if some inter-body distance tends to infinity, meaning that some body or cluster of bodies "escapes to infinity''.

Density Conjecture: In arbitrarily close proximity to the initial conditions for any bounded solution, lies an initial condition whose solution is unbounded.

In section 7 of his 1998 invited lecture at the International Congress of Mathematicians, Michael Herman brought wide attention to the Density conjecture, and called it "The Oldest Open Problem in Dynamical Systems". He asserted that Newton "certainly believed" the conjecture, having invoked God as the source of control for the instabilities of the N-body problem. For further detail on the problem see section 14.2 of this reference

Like many simply stated problems in mathematics, the Proximity Conjecture has proved maddeningly difficult to assess. In Christian Marchal's influential book on the three-body problem, he assumes fairly explicitly, but without proof, that the answer is true, essentially appealing to the idea that given sufficient time, "everything that can happen, will happen". The KAM theorem moreover, asserts that for every "good periodic" solution, there is a set of positive measure of solutions which stay close to that solution for all time, and hence are bounded. These solutions form the KAM torii. There exist, however, lots of "holes" in the torii. Arnol'd diffusion is a class of mechanisms, exploiting resonances, by which one can "wander" from hole to hole and thereby eventually escape to infinity. So far, the main approach to proving the Density Conjecture has involved efforts to show that Arnol'd diffusion is ubiquitous.

Will the Density Conjecture be proved true for the planar 3-body problem in the next 10 years?

Resolution is positive if a proof of a theorem to which the above description applies with reasonable accuracy is published by Sept. 1, 2027. Additionally, if the conjecture is proved for the planar three body problem with particular (all nonzero) mass ratios, resolution will also be positive. Finally, the question resolves negative if a negative proof or counterexample is found prior to the resolution date, or if no proof is published at all.

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