  In this paper we study the existence of periodic orbits with prescribed
energy levels of convex Lagrangian systems on complete Riemannian manifolds. We
extend the existence results of Contreras by developing a modified minimax
principal to a class of Lagrangian systems on noncompact Riemannian manifolds,
namely the so called $\lsh$ Lagrangian systems. In particular, we prove that
for almost every $k\in(0,c_u(L))$ the exact magnetic flow associated to a
$\lsh$ Lagrangian has a contractible periodic orbit with energy $k$. We also
discuss the existence and non-existence of closed geodesics on the product
Riemannian manifold $\R\times M$.
