We prove that the Hamiltonian system

$$
\begin{cases} 
\.{p}\: = \:−\frac{∂\mathrm{V}}{∂q} \\ 
\.{q}\: = \:p  
\end{cases}
\qquad p,\:q \in \mathrm{R}^{n};\ \mathrm{V} \in \mathrm{C}^{2}\left(\mathrm{R}^{n}\right)
$$

has at least one periodic solution of energy $h$, provided that the set $\{q \in \mathrm R^n | \mathrm V(q) ⩽ h\}$ is compact.


### Résumé

Nous démontrons que le système hamiltonien

$$
\begin{cases} 
\.{p}\: = \:−\frac{∂\mathrm{V}}{∂q} \\ 
\.{q}\: = \:p  
\end{cases}
\qquad p,\:q \in \mathrm{R}^{n};\ \mathrm{V} \in \mathrm{C}^{2}\left(\mathrm{R}^{n}\right)
$$

a au moins une solution périodique d’énergie $h$, pourvu que l’ensemble $\{q \in \mathrm R^n | \mathrm V(q) ⩽ h\}$ soit compact.


Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems

V. Benci

Abstract

Résumé

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