Mathematical Research Letters

Volume 19 (2012)

Number 2

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

Pages: 483 – 497



Kuo-Chang Chen (Department of Mathematics and National Center for Theoretic Sciences, National Tsing Hua University, Hsinchu 300, Taiwan.)

Tiancheng Ouyang (Department of Mathematics, Brigham Young University, Provo, Utah 84602, U.S.A.)

Zhihong Xia (Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A.; CEMA, Central University of Finance and Economics, Beijing, China.)


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.

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