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

Abstract

Abstract. Considering any set of n-positive masses, n ≥ 3, moving in R 2 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 EulerMoulton 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.

Simple Choreographies of the Planar Newtonian N-Body Problem

“…The Figure-Eight solution was originally proved in [13] using this method. Results obtained using this method can also be found in [2], [3], [4], [6], [7], [5] and the references within. Despite of the above progresses, up to our knowledge very few simple choreographies have been rigorously proved.…”

“…First by the symmetric constraints, z = (z j ) j∈N ∈ Λ DN N must satisfy (4) and (7) z j (t) =z N −1−j (1 − t), ∀t ∈ R, ∀j ∈ N.…”

Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem

“…(25) imply that A kj = 0 for k = 1, 2, 3, 4 and j = 1, 2. Because the relations (19) is equivalent to A kj = 0, we complete the proof that q(t) connects very well at t = 2T . Now we prove that there exists a minimizing path which is different from the circular motion.…”

“…Many expertises attempt to study choreographic solutions and a large number of simple choreographic solutions have been discovered numerically but very few of them have rigorous existence proofs. More results can be found in [14][15][16]10,[17][18][19][20] and the reference therein.…”

“…This idea of SPBC had been reported by one of the authors, Tiancheng Ouyang in the HAMSYS conference, March 2001 (see Chenciner's Remark in [26]). Some preliminary results were included in the paper [19] which was submitted in 2003 and was finally published in 2012. The method works for general n-body problem without any constraints on masses or symmetries.…”