Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds

We study the existence of periodic solutions of singular Hamiltonian systems as well as closed geodesics on non-compact Riemannian manifolds via variational methods. For Hamiltonian systems, we show the existence of a periodic solution of prescribed-energy problem: q̇̇+∇V(q)=0, 1/2|q̇|²+V(q)=0 under the conditions: (i) V(q) < 0 for all q ∈ ℝ^N \ {0}; (ii) V(q) ∼ -1/|q|² as |q| ∼ 0 and \q\ ~∼ ∞. For closed geodesics, we show the existence of a non-constant closed geodesic on (ℝ × S^N-1, g) under the condition: g_(s,x) ∼ ds² + h₀ as s ∼ ± ∞, where h₀ is the standard metric on S^N-1.