Abstract
Abstract 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 θ ∈ ( 0 , π / 7 ] {\theta\in(0,\pi/7]} in the equal-mass four-body problem. A new geometric argument is introduced to show that for any θ ∈ ( 0 , π / 2 ) {\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.
Highlights
In 2003, Vanderbei [13] successfully applied his optimizing program to the N-body problem and found many periodic orbits numerically
There is an interesting class of orbits in the parallelogram four-body problem with equal-masses, which he named as double-double orbits
Double-double orbit is called retrograde if one pair of the two adjacent masses revolves around each other in one direction while their mass center revolves around the mass center of the other pair in a different direction
Summary
In 2003, Vanderbei [13] successfully applied his optimizing program to the N-body problem and found many periodic orbits numerically. The retrograde double-double orbits can be seen as one of their many applications They ([4, 9]) showed that in the parallelogram equal-mass four-body problem, there exists a set of collision-free action minimizer connecting a collinear configuration and a rectangular configuration. This set coincides with the set of retrograde double-double orbits. A direct application is to analyze the properties of the retrograde orbits and the prograde orbits with mass M = [1, 1, m] It is known [6, 7] that, for each given θ ∈ (0, π/2), both orbits can be characterized as action minimizers connecting a collinear configuration and an isosceles configuration.
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