# Can we infer the existence of periodic solutions to the three-body problem from numerical evidence?

It seems like they were able to rigorously prove the existence of N-body choreographies by using interval Krawczyk method to show that a minimum exist to the variational problem solved in the subspace of the full phase space satisfying some symmetry conditions.

Following the links given I found this paper where they explain the method. It's not exactly a light reading material but on page 6 they say: "If all these conditions all fulfilled, then from Theorem 4.5 we are sure that in the set $Z \times \{c_0\}$ there is an initial condition for the choreography. Moreover, as the set Z is usually very small, the shape of the proved choreography is very similar to our first approximation."

It sounds like starting with "an initial guess", they are able to show that there exist an "exact solution" very close to this initial guess. And one can probably obtain a curve that is arbitrarily close to the actual solution by doing more and more precise calculation. But the existence of the choreography is established rigorously with the assist of their numerical method.

Note that in the beginning of the paper, they mention the solutions obtained by the usual numerical methods as "solutions produced in a non-rigorous numerical way."