Existence of Prograde Double-Double Orbits in the Equal-Mass Four-Body Problem

By introducing simple topological constraints and applying a binary decomposition method, we show the existence of a set of prograde double-double orbits for any rotation angle {\theta\in(0,\pi/7]} in the equal-mass four-body problem. A new geometric argument is introduced to show that for any {\theta\in(0,\pi/2)}, the action of the minimizer corresponding to the prograde double-double orbit is strictly greater than the action of the minimizer corresponding to the retrograde double-double orbit. This geometric argument can also be applied to study orbits in the planar three-body problem, such as the retrograde orbits, the prograde orbits, the Schubart orbit and the Hénon orbit.