Lyapunov center theorem on rotating periodic orbits for Hamiltonian systems.

We introduce the Q (s) -index ind Γ for a symplectic orthogonal group Q (s) and Q (s) invariant subset Γ of R 2 n and prove that ind S 2 n − 1 = n. Using this fact, we study multiple rotating periodic orbits of Hamiltonian systems. For an orthogonal matrix Q , a Q -rotating periodic solution z (t) has the form z (t + T) = Q z (t) for all t ∈ R and some constant T > 0. According to the structure of Q , it can be periodic, anti-periodic, subharmonic, or just a quasi-periodic one. Under a non-resonant condition, we prove that on each energy surface near the equilibrium, the Hamiltonian system admits at least n Q -rotating periodic orbits, which can be regarded as a Lyapunov type theorem on rotating periodic orbits. [ABSTRACT FROM AUTHOR]