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.Comment: Total 26 pages including 5 pages of figures and references, 12 figure