Families of symmetric relative periodic orbits originating from the circular Euler solution in the isosceles three-body problem

Abstract

We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches. We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge to "previously unknown" periodic orbits in the planar problem.

Bifurcations from one-parameter families of symmetric periodic orbits in reversible systems

“…This fact was repeatedly used in [18,19] although it was not so clearly stated there. A homoclinic version of this result was also used as the key idea in [25].…”

“…Here the boundary values ξ 1 (0), ξ 1 (T ), η 1 (0) and η 1 (T ) were also taken as free parameters and the Hamiltonian energy H was monitored. A similar numerical approach was also used for computing symmetric relative periodic orbits in the isosceles three-body problem [19] equations [13] to obtain general results.…”

“…Remark 4.5. Recently, Maier [12] gave an efficient method to compute the monodromy matrix of the Lamé equation (19) for ℓ ∈ Z. His method can be used for necessary computations when Theorem 3.6 is applied to the case of ℓ ∈ Z in (14).…”

Periodic orbit in the frame work of restricted three bodies under the asteroids belt effect

“…Moreover he use Levi-Civita transformation to regularize the equations of motion, in order to avoid the singularity between the third body and one of the primary bodies. [32] used theoretical and numerical approaches to investigate and study the symmetric relative periodic orbits within frame of the isosceles restricted problem three bodies. They also proved that the elastance of many families of symmetric relative periodic solution, which are emerged from heteroclinic connections between binary or triple collisions [14] studied the real system of Saturn-Titan to explore the oblateness influence of Saturn planet on the periodic orbits and quasi-periodic motion regions around the primaries within frame restricted thee-body model.…”

Reduction the secular solution to periodic solution in the generalized restricted three-body problem

“…Furthermore, he regularized the equations of motion of the problem using the Levi-Civita transformations to avoid the singularity due to binary collisions between the third body and one of the primaries. Symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches are studied by Shibayama and Yagasaki (2011). They proved that another family of symmetric relative periodic orbits is born from the circular Euler solution besides the elliptic Euler solutions.…”