UC Santa Cruz

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 figure eight orbit discovered by C. Moore [ Moore, C.: Phys. Rev. Lett. 70, 3675 - 3679 ( 1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881 - 901 ( 2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates with frequency Omega around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93 - 99 ( 2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron. 78, 279 - 298 ( 2001)], and more recently in [ Chenciner, A. et al.: Nonlinearity 18, 1407 - 1424 ( 2005)] 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 Omega which varies between Omega = 0, for the planar figure eight orbit with intrinsic frequency omega, and Omega = omega for the 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 [ Nauenberg, M.: Phys. Lett. 292, 93 - 99 ( 2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given numerically, and some examples are given of nearby stable orbits which bifurcate from these families.