Shape space figure-8 solution of three body problem with two equal masses

Abstract

In a preprint by Montgomery [12], the author attempted to prove the existence of a shape space Figure-8 solution of the Newtonian three body problem with two equal masses (it looks like a figure 8 in the shape space, which is different from the famous Figure-8 solution with three equal masses [6]). Unfortunately there is an error in the proof and the problem is still open.Consider the α-homogeneous Newton-type potential, 1/r α , using action minimization method, we prove the existence of this solution, for α ∈ (1, 2); for α = 1 (the Newtonian potential), an extra condition is required, which unfortunately seems hard to verify at this moment.

Spatial Double Choreographies of the Newtonian 2n-Body Problem

“…A detailed proof of this lemma in the planar case can be found in [32,Proposition 4.3]. The spatial case can be proven similarly.…”

The Broucke–Hénon orbit and the Schubart orbit in the planar three-body problem with two equal masses

“…Let r i j = q i − q j . In the three-body problem, it is known [34] that lim t→0 + r i j (t) |r i j (t)| exists if a binary collision between m i and m j occurs at t = 0. We define the direction of collision c i j to be…”

“…By Corollary 4.5, the direction of collision c 12 = lim t→0 + q 1 −q 2 |q 1 −q 2 | must be (1, 0). By Yu's work [34], more information can be shown about the colliding pair: bodies 1 and 2. After taking away the movement of the center of mass of this colliding pair, the asymptotic behavior of q 1 − q 2 at t = 0 satisfies: Lemma 5.2.…”

“…Ifq 1y (0) =q 2y (0) = 0, by Lemma 6.2 in [34], the motion must be collinear. In this case, the minimizer P coincides with the Schubart orbit.…”

Simple Choreographies of the Planar Newtonian N-Body Problem

“…• Local deformation: assuming there is an isolated collision along the minimizing path, one tries to show that after a small deformation near the isolated collision, one gets a new path with strictly smaller action value, which gives a contradiction. For the details see [22], [27], [29], [19], [8] and [14]. In the last three references the existence of such a local deformation was implied implicitly through Marchal's average method.…”