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Minimax periodic orbits of convex Lagrangian systems on complete Riemannian manifolds

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 $\R\times M$.