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). 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.
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