Lyapunov center theorem on rotating periodic orbits for Hamiltonian systems

Abstract

We introduce the Q(s)-index ind Γ for a symplectic orthogonal group Q(s) and Q(s) invariant subset Γ of R2n and prove that indS2n−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)=Qz(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.