Periodic orbits and subharmonics of dynamical systems on non-compact Riemannian manifolds (Q1924452)

The authors consider a system on a Riemannian manifold \(M\) defined by a potential function \(V:M\to\mathbb{R}\): \[ D_t(x(t))= -\nabla V(x(t))\tag{1} \] where \(D_t\) is the covariant derivative, \(\nabla\) is the gradient. The aim of the paper is to prove the existence of periodic solutions of (1), when \(M\) is a complete, connected, finite-dimensional manifold. For autonomous \(V(x)\) conditions for the existence of periodic solutions with a prescribed period are given. For \(V(x,t)\) periodic in \(t\) conditions for the existence of subharmonics are given. Proofs are based on the fact that the search for periodic solutions of (1) can be reduced to the search for critical points of the action functional.