Symmetric regularization, reduction and blow-up of the planar three-body problem

Abstract

PROOFS - PAGE NUMBERS ARE TEMPORARY PACIFIC JOURNAL OF MATHEMATICS Vol. , No. , 2013 dx.doi.org/10.2140/pjm.2013..101 SYMMETRIC REGULARIZATION, REDUCTION AND BLOW-UP OF THE PLANAR THREE-BODY PROBLEM R ICHARD M OECKEL AND R ICHARD M ONTGOMERY We carry out a sequence of coordinate changes for the planar three-body problem, which successively eliminate the translation and rotation symme- tries, regularize all three double collision singularities and blow-up the triple collision. Parametrizing the configurations by the three relative position vectors maintains the symmetry among the masses and simplifies the regu- larization of binary collisions. Using size and shape coordinates facilitates the reduction by rotations and the blow-up of triple collision while empha- sizing the role of the shape sphere. By using homogeneous coordinates to describe Hamiltonian systems whose configurations spaces are spheres or projective spaces, we are able to take a modern, global approach to these familiar problems. We also show how to obtain the reduced and regularized differential equations in several convenient local coordinates systems. 1. Introduction and history The three-body problem of Newton has symmetries and singularities. The reduction process eliminates symmetries thereby reducing the number of degrees of freedom. The Levi-Civita regularization eliminates binary collision singularities by a nonin- vertible coordinate change together with a time reparametrization. The McGehee blow-up eliminates the triple collision singularity by an ingenious polar coordi- nate change and another time reparametrization. Each process has been applied individually and in various combinations to the three-body problem, many times. In this paper we apply all three processes globally and systematically, with no one body singled out in the various transformations. The end result is a complete flow on a five-dimensional manifold with boundary. We focus attention on the geometry of the various spaces and maps appearing along the way. At the heart of this paper is a beautiful degree-4 octahedral covering map of the shape sphere, branched over the binary collision points (see Figure 4 on page 151). This map Research supported by NSF grant DMS-1208908. MSC2010: primary 37N05, 70F07, 70G45; secondary 53A20, 53CXX. Keywords: celestial mechanics, three-body problem, regularization.