On Action-Minimizing Retrograde and Prograde Orbits of the Three-Body Problem

Abstract

A retrograde orbit of the planar three-body problem is a relative periodic solution with two adjacent masses revolving around each other in one direction while their mass center revolves around the third mass in the other direction. The orbit is said to be prograde or direct if both revolutions follow the same direction. Let T > 0 and ${\phi\in[0,2\pi)}$ be fixed, and consider the rotating frame which rotates the inertia frame about the origin with angular velocity ${\frac{\phi}{T}}$ . In a recent work of K.-C.Chen [5], the existence of action-minimizing retrograde orbits which are T-periodic on this rotation frame were proved to exist for a large class of masses and a continuum of ${\phi}$ . In this paper we generalize the main result in [5], provide some quantitative estimates for admissible masses and mutual distances, and show miscellaneous examples of action-minimizing retrograde orbits. We also show the existence of some prograde and retrograde solutions with additional symmetries.