Normal modes of a Lagrangian system constrained in a potential well

V. Benci

doi:10.1016/s0294-1449(16)30419-x

JournalsaihpcVol. 1, No. 5pp. 379–400

Normal modes of a Lagrangian system constrained in a potential well

V. Benci

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Abstract

Let $a, U \in C^{2} (Ω)$ where $Ω$ is a bounded set in $R^{n}$ and let

L (x, ξ) = \frac{1}{2} a (x) ∣ ξ ∣^{2} - U (x), x \in Ω; ξ \in R^{n} .

We suppose that $a, U > 0$ for $x \in Ω$ and that

x \to \partial Ω lim U (x) = + \infty.

Under some smoothness assumptions, we prove that the Lagrangian system associated with the above Lagrangian $L$ has infinitely many periodic solutions of any period $T$ .

Résumé

Soit $a, U \in C^{2} (Ω)$ où $Ω$ est un borné de $R^{n}$ , on pose

L (x, ξ) = \frac{1}{2} a (x) ∣ ξ ∣^{2} - U (x), x \in Ω; ξ \in R^{n} .

Nous supposons que $a, U > 0$ pour $x \in Ω$ et que

x \to \partial Ω lim U (x) = + \infty.

Moyennant quelques hypothèses de différentiabilité, nous démontrons que le système Lagrangien associé à $L$ a une infinité de solutions $T$ -périodiques, quel que soit $T$ .

Cite this article

V. Benci, Normal modes of a Lagrangian system constrained in a potential well. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 5, pp. 379–400

DOI 10.1016/S0294-1449(16)30419-X