A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem

Abstract

After the existence proof of the first remarkably stable simple choreographic motion-- the figure eight of the planar three-body problem by Chenciner and Montgomery in 2000, a great number of simple choreographic solutions have been discovered numerically but very few of them have rigorous existence proofs and none of them are stable. Most important to astronomy are stable periodic solutions which might actually be seen in some stellar system. A question for simple choreographic solutions on $n$-body problems naturally arises: Are there any other stable simple choreographic solutions except the figure eight? In this paper, we prove the existence of infinitely many simple choreographic solutions in the classical Newtonian 4-body problem by developing a new variational method with structural prescribed boundary conditions (SPBC). Surprisingly, a family of choreographic orbits of this type are all linearly stable. Among the many stable simple choreographic orbits, the most extraordinary one is the stable star pentagon choreographic solution. The star pentagon is assembled out of four pieces of curves which are obtained by minimizing the Lagrangian action functional over the SPBC. We also prove the existence of infinitely many double choreographic periodic solutions, infinitely many non-choreographic periodic solutions and uncountably many quasi-periodic solutions. Each type of periodic solutions have many stable solutions and possibly infinitely many stable solutions.

arXiv commentTotal 26 pages including 5 pages of figures and references, 12 figures

References in corpus (5)

• On the Existence of Collisionless Equivariant Minimizers for the Classical n-body Problem
• Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry
• Action minimizing orbits in the n-body problem with simple choreography constraint
• Unchained polygons and the N-body problem
• New Orbits for the n-Body Problem

Cited by in corpus (4)

• The Broucke-Hénon orbit and the Schubart Orbit in the planar three-body problem with equal masses
• Action minimizers under topological constraints in the planar equal-mass four-body problem
• Stable motions of high energy particles interacting via a repelling potential
• Geometric properties of minimizers in the planar three-body problem with two equal masses