Abstract
The existence of periodic solutions of the N-body problem has been widely studied. The case n=2, called the Keplerian problem, is well-understood. When n is equal or bigger than 3, this problem has stimulated a series of research about the N-body problem. Euler(1767) and Lagrange(1772) made some fundamental contributions. In 2000 [5], Chenciner and Montgomery used calculus of variations to prove the existence of the figure-8 orbit of the three-body problem. After this, the existence of many N-body problems were obtained. Because the method used is calculus of variations on symmetric path spaces, it relies on some equal-mass conditions. Therefore, the problems solved are confined to certain cases with symmetry assumptions. In 2006 [1], professor K.-C Chen proved the existence of retrograde orbits of the three-body problem. This result is a significant breakthrough regarding the existence of solutions of the three-body problem. In addition to the existence for the N-body problem, we are also interested in some quantitative properties. In this article, we obtain upper bound estimates for mutual distances of action-minimizing retrograde orbits of the three-body problem. In section 1, we briefly introduce the N-body problem, some basic equations and notations. In section 2, we explain the existence of retrograde orbits of the three-body problem. This part of discussion mainly comes from [1]. Section 3 contains main results of this article. We use symmetry, the triangle inequality, and the method of iterations to find out the upper bound estimates for the mutual distances of particle. In section 4, we use some numerical method to calculate these upper bound estimates.