Action-Minimizing Periodic and Quasi-Periodic Solutions in the $n$-body Problem

Considering any set of n-positive masses, n ≥ 3, moving in R2 under Newtonian gravitation, we prove that action-minimizing solutions in the class of paths with rotational and reflection symmetries are collision-free. For an open set of masses, the periodic and quasi-periodic solutions we obtained contain and extend the classical Euler– Moulton relative equilibria. We also show several numerical results on these actionminimizing solutions. Using a natural topological classification for collision-free paths via their braid types in a rotating frame, these action-minimizing solutions change from trivial to non-trivial braids as we vary masses and other parameters.