Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit

In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent discoveries in the n-body problem. The key ingredient is a generalized Bott-type iteration formula for periodic solution in the presence of finite group action on the orbit. For second order system, we prove, under general boundary conditions, the close formula for the relationship between the Morse index of an orbit in a Lagrangian system and the Maslov index of the fundamental solution for the corresponding orbit in its Hamiltonian system counterpart, and the boundary conditions cover the cases which appeared in the n-body problem. As an application we consider the stability problem of the celebrated figure-eight orbit due to Chenciner and Montgomery in the planar three-body problem with equal masses, and we clarify the relationship between linear stability and its variational nature on various loop spaces. The basic idea is as follows: the variational characterization of the figure-eight orbit provides information about its Morse index; based on its relation to the Maslov index, our stability criteria come into play. Partially supported by NSFC (No.10801127) and the knowledge innovation program of the Chinese Academy of Science. Partially supported by NSFC (No.s 10401025, 10571123 and 10731080) and NSFB-FBEC (No. KZ20 0610028015).