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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 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.

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    $Y$ou may be interested in [this paper](http://www.jstor.org/stable/2$0$46321), the references therein, and so forth. Leave this question here for a while. If you don't get better answers, you should consider asking on MathOverflow.2012-02-10

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