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# Mathematical Research Letters

## Volume 19 (2012)

### Number 2

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

Pages: 483 – 497

DOI: http://dx.doi.org/10.4310/MRL.2012.v19.n2.a19

#### Authors

#### Abstract

Considering any set of $n$-positive masses, $n \geq 3$, moving in$\mathbb{R}^2$ under Newtonian gravitation, we provethat action-minimizing solutions in the class of paths withrotational and reflection symmetries are collision-free.For an open set of masses, the periodic and quasi-periodic solutions we obtainedcontain and extend the classical Euler--Moulton relative equilibria.We also show several numerical results on these action-minimizing solutions.Using a natural topological classification for collision-free paths viatheir braid types in a rotating frame, these action-minimizing solutionschange from trivial to non-trivial braids as we vary masses and other parameters.