Symmetry Groups and Non-Planar Collisionless Action-Minimizing Solutions of the Three-Body Problem in Three-Dimensional Space

Abstract

Periodic and quasi-periodic solutions of the n-body problem can be found as minimizers of the Lagrangian action functional restricted to suitable spaces of symmetric paths. The main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three-body problem in the three-dimensional space. First we give a finite complete list of symmetry groups fit to the minimization of the action, with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives. A second step is to prove that the resulting (local and global) symmetric action-minimizers are always collisionless (when they are not already bound to collisions). Furthermore, we prove some results addressed to the question whether minimizers are planar or non-planar; as a consequence of the theory we will give general criteria for a symmetry group to yield planar or homographic minimizers (either homographic or not, as in the Chenciner-Montgomery eight solution); on the other hand we will provide a rigorous proof of the existence of some interesting one-parameter families of periodic and quasi-periodic nonplanar orbits. These include the choreographic Marchal's P 12 family with equal masses -together with a less-symmetric choreographic family (which anyway probably coincides with the P 12 ).MSC Subj. Class: Primary 70F10; Secondary 70G75, 37C80, 70F16, 37J45.

Symmetries and choreographies in families that bifurcate from the polygonal relative equilibrium of the n-body problem

“…In [13], Terracini and Ferrario (2004) applied the principle of least action systematically over symmetric paths to avoid collisions, using ideas introduced by Marchal [21]. For the discussion of these and other variational approaches we refer to [2,3,11,12,28], and references therein.…”

Spatial Double Choreographies of the Newtonian 2n-Body Problem

“…Since then, many new periodic and quasi-periodic solutions have been found using this method. It is impossible to give a complete list, to name a few of them, see [1], [2], [17], [16], [25] and the references with in .…”

“…In general, when we are considering a minimization problem in R 3 , it is not an easy task to determine whether an action minimizer is planar or spatial, see [3]. For example, the action of the dihedral group D 6 of the Figure-Eight of the planar three-body problem can be extended to R 3 , however so far no proof is available regarding whether the corresponding action minimizer is planar or spatial, see [12] or [16]. The same problem also occurs if we try to extend the minimization problem considered in [33] to R 3 .…”

On the variational approach to the periodic n-body problem

“…Based on the classification of planar groups, by introducing a natural notion of space extension of a planar group, Ferrario (2004) gave a complete answer to the classification problem for the three-body problem in the space and at the same time to determine the resulting minimizers and describe its more relevant properties.…”