Hyperbolicity for symmetric periodic orbits in the isosceles three body problem

Abstract

We study the isosceles three body problem with fixed symmetry line for arbitrary masses, as a subsystem of the N-body problem. Our goal is to construct minimizing noncollision periodic orbits using a symmetric variational method with a finite order symmetry group. The solution of this variational problem gives existence of noncollision periodic orbits which realize certain symbolic sequences of rotations and oscillations in the isosceles three body problem for any choice of the mass ratio. The Maslov index for these periodic orbits is used to prove the main result, Theorem 4.1, which states that the minimizing curves in the three dimensional reduced energy momentum surface can be extended to periodic curves which are generically hyperbolic. This reminds one of a theorem of Poincaré [8], concerning minimizing periodic geodesics on orientable 2D surfaces. The results in this paper are novel in two directions: in addition to the higher dimensional setting, the minimization in the current problem is over a symmetry class, rather than a loop space.

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

“…(ii) Extending the results of Offin and Cabral (2009) and Shibayama (2009a), we can show that an R 1 -and R 2 -symmetric periodic orbit with any period exists in (7) even if α = 1 3 . See Appendix A.…”

“…Offin and Cabral (2009) and Shibayama (2009a) independently showed the existence of relative periodic orbits in the isosceles three-body problem corresponding to only R 2 -symmetric periodic orbits.…”

“…3. A generalization of the results in Cabral (2009) andShibayama (2009a) is also presented in Appendix A. Finally we give numerical results for symmetric periodic orbits in the blown-up system and symmetric relative periodic orbits in the isosceles threebody problem in Sect.…”

“…On the other hand, Offin and Cabral (2009) and Shibayama (2009a) independently showed the existence of a family of symmetric relative periodic orbits except the Euler solutions using variational methods. Moreover, Offin and Cabral (2009) proved the instability of the family while Shibayama (2009a) proved the stability of the Euler solutions with small eccentricity.…”

“…Moreover, Offin and Cabral (2009) proved the instability of the family while Shibayama (2009a) proved the stability of the Euler solutions with small eccentricity. Shibayama (2011) also showed the existence of planar periodic orbits with binary collisions for any period.…”