Continuity and Stability of families of figure eight orbits with finite angular momentum

Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects the classical circular orbit of Lagrange with the recently discovered planar figure eight orbit with zero total angular momentum [1],[2]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped orbits with finite angular momentum orbits were first reported in [3], and mathematical proofs for the existence of such orbits were given in [5], and more recently in [6] where also some numerical solutions have been presented. Numerical evidence is given here that the family of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0 ( for the planar figure eight orbit with intrinsic frequency ω), and Ω = ω (for a circular Lagrange orbit) Similar numerical solutions are also found for n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the figure eight orbit [3], and some new results are given here. Recently, existence proofs for families of such orbit and further numerical solutions have been given in [6]. The stability of these orbits is examined numerically without the restriction to a linear approximation [6], and some examples are given of nearby stable orbits which bifurcate from these families.