JournalsaihpcVol. 1, No. 5pp. 401–412

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

  • V. Benci

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

We prove that the Hamiltonian system

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

has at least one periodic solution of energy h, provided that the set {q ∈ Rn|V(q) ⩽ h} is compact.

Résumé

Nous démontrons que le système hamiltonien

\begin{matrix} p\limits^{˙}\: = \:−\frac{∂\mathrm{V}}{∂q} \\ q\limits^{˙}\: = \:p \\ \end{matrix}\:\:\:\:\:\:\:\:\:\:\: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 ∈ Rn|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