Let be given a mechanical system with configuration manifold $M,$ potential energy $V$ and kinetic energy $K$ corresponding to a riemannian metric on $M.$ Its dynamics is determined by the Euler-Lagrange vector field on $TM$ corresponding to $L=K-V.$
For a given value $e$ of the total energy $K+V,$ let us denote by $n(e)$ the number of closed trajectories of motion with energy, (here I am identifying motions differing just by a time translation).
Question.1 Under what hypothesis on the whole mechanical system, or only on the value $e$, is it possible to give a positive lower bound on $n(e)$?
I was motivated even for having been puzzled, probably for lacking preparation, by a statement in §4.2.1 in Arnol'd, Kozlov, Neishtadt Mathematical Aspects of Classical and
Celestial Mechanics, (3rd Edn).
There I found that, invoking Hadamard(1898) (in any homotopy free class of a not simply connected riemannian manifold there exists a closed geodesic) and Maupertuis' principle, when $M$ is not simply connected and $\sup_M V<e,$ it is possile to extimate from below $n(e).$
Question.2 What kind of lower bound they are referring to? Probably it is elementary, and I am missing something, but what?
Excuse me if the question is not well-posed, any suggestion in order to improve the terms of the problem are welcome, just as the answers of course.