Classification of periodic orbits in the four- and five-body problems.

Abstract

The research described in this paper was motivated by the new types of periodic solutions that were recently discovered by Moore, Chenciner, Montgomery, and Simo in the three-body and the N-body problem (with large N). We attempt to classify the various types of periodic orbits, in the inertial frame, on the basis of an extensive numerical exploration. We have started an exploration of the four-body problem, where the classification of types of periodic orbits is more involved. We immediately found those orbits with symmetry with respect to the x-axis, the y-axis, or both. Complete quadruple interplay, or a triple system around a single mass is possible. Finally, what seems to be the most frequent, two binary systems in orbit around the general center of mass. However, the last group can be separated into several subgroups. We have, for instance, the case of two masses on one orbit and the two other masses on another orbit. This is again a partial choreography. We then have the case of double binary choreographies: two binary systems in which all four masses travel on one single curve. To realize this situation, a certain commensurability is needed between the overall period and the period of each binary system. The binary system makes an odd integer number, q, of revolutions during a general period. We computed many cases from q = 5 to 97. This indicates that we have a discrete infinity of choreographies. The choreography q = 5 is an especially remarkable star-shaped curve. Actually, we have two such infinite families, one with corotational and the other with contrarotational motions. We show that they can be justified by simple Keplerian approximations. Finally, we mention that our work depends heavily on the existence of symmetries. They simplify the algorithms for finding periodic orbits and they play an important role in the classification. In the four-body problem with equal masses, we have five different types of symmetries. In addition to the well-known collinear symmetry, we also found what we call double isosceles and trapezoidal symmetries. All our four-body choreographies satisfy these two symmetry conditions. In the five-body problem, we limit our work to two symmetry cases. We show that in Gerver's super-eight choreography, we are allowed to place a fifth mass at the center and we may speak about a larger sun with four smaller but equal planets that are in a choreography.