Bifurcation of Relative Equilibria of the (1+3)-Body Problem

Abstract

Abstract. We study the relative equilibria of the limit case of the planar Newtonian 4-body problem when three masses tend to zero, the so-called (1+3)-body problem. Depending on the values of the infinitesimal masses the number of relative equilibria varies from ten to fourteen. Always six of these relative equilibria are convex and the others are concave. Each convex relative equilibrium of the (1+3)-body problem can be continued to a unique family of relative equilibria of the general 4-body problem when three of the masses are sufficiently small and every convex relative equilibrium for these masses belongs to one of these six families.

Classifying Four-Body Convex Central Configurations

“…We will find a set D ⊂ R + 3 such that for each (a, b, c) ∈ D, there exists a unique angle θ which makes the configuration central. Specifically, we prove that there exists a differentiable function θ = f (a, b, c) with domain D, whose graph is equivalent to E. In order to define D, we use the mutual distance inequalities in (9) to eliminate the angular variable θ.…”

“…Configurations on face IV or V, respectively, correspond to equilibria of the planar, circular, restricted four-body problem with infinitesimal mass m 3 or m 4 , respectively [7,8,20]. Configurations on face III or VI, respectively, correspond to relative equilibria of the (1 + 3)-body problem, where a central mass (body 1 or 2, respectively) is equidistant from three infinitesimal masses [9,16,26]. Note that we have not made any assumptions on the relative size of the masses.…”

“…which is non-negative due to the inequalities in (9). Note that the quantity β 2 is identical to α 3 used in the proof of Theorem 3.6 (equation ( 40)).…”

“…which is strictly negative due to the inequalities in (9). Finally, we compute that Using equations ( 13) and ( 14), we have that Then we compute…”

On Centered Co-circular Central Configurations of the n-Body Problem

“…The difficulty of the problem of finding central configurations of the general n-body problem forces us to consider some simplifications by imposing restrictions, usually on the masses or the geometry of the configuration. The most common simplifications are to take some equal masses or some infinitesimal masses and to impose symmetries or a fixed shape on the configuration, see for instance [1,20,27,3,33,8,16,9,10] and the reference therein.…”