The Application and Proof of Group Theory in the Three-body Problem

There are various symmetries of great importance in modern mathematical physics. These symmetries can enormously reduce computation and simplify the solution of many complicated dynamic systems and differential equations. Among the applications of the group theory, the applications to solving dynamic equations of the planar three-body problem is a study worthy case. The aim of this paper is to illustrate the application of group action to the three-body problem. The author introduced the work of Chenciner and Montgomery published on the Annual of Mathematics in 2000. In this paper, the author illustrates the proof sketch of the main Lemma in the paper of A.C. and R.M. and analyzes the main steps concerning group action. Two applications of group were demonstrated in this paper. The first one is reducing the solution space of the problem with the symmetry group of the sphere, SO(2). The second one is the action of the Klein group, used to generate the entire orbit of the solution from a piece of loop. Finally, the potential applications of group theory to more dynamical problems.