Adiabatic limits of closed orbits for some Newtonian systems in ${\mathbb{R}}$ n

Abstract

We deal with a Newtonian system like $\ddot{x}$ +V′(x)=0. We suppose that V : ${\mathbb{R}}$ ⁿ→ ${\mathbb{R}}$ possesses an (n−1)‐dimensional compact manifold M of critical points, and we prove the existence of arbitrarily slow periodic orbits. When the period tends to infinity these orbits, rescaled in time, converge to some closed geodesics on M.