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

V. Benci

doi:10.1016/s0294-1449(16)30420-6

JournalsaihpcVol. 1, No. 5pp. 401–412

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

V. Benci

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Abstract

We prove that the Hamiltonian system

{\overset{p}{˙} = - \frac{\partial V}{\partial q} \overset{q}{˙} = p p, q \in R^{n}; V \in C^{2} (R^{n})

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

Résumé

Nous démontrons que le système hamiltonien

{\overset{p}{˙} = - \frac{\partial V}{\partial q} \overset{q}{˙} = p p, q \in R^{n}; V \in C^{2} (R^{n})

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

Cite this article

V. Benci, Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 5, pp. 401–412

DOI 10.1016/S0294-1449(16)30420-6