JournalsaihpcVol. 1, No. 5pp. 379–400

Normal modes of a Lagrangian system constrained in a potential well

  • V. Benci

Normal modes of a Lagrangian system constrained in a potential well cover
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Abstract

Let a, U ∈ C2(Ω) where Ω is a bounded set in ℝn and let

L(x,ξ)=12a(x)ξ2U(x),xΩ;ξRn.\mathrm{L}\left(x,\xi \right) = \frac{1}{2}a\left(x\right)\left|\xi \right|^{2}−\mathrm{U}\left(x\right),\:\:x \in Ω;\xi \in ℝ^{n}.

We suppose that a, U > 0 for x ∈ Ω and that

limxΩU(x)=+.\lim \limits_{x\rightarrow ∂Ω}\mathrm{U}\left(x\right) = + ∞.

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 ∈ C2(Ω) où Ω est un borné de ℝn, on pose

L(x,ξ)=12a(x)ξ2U(x),xΩ;ξRn.\mathrm{L}\left(x,\xi \right) = \frac{1}{2}a\left(x\right)\left|\xi \right|^{2}−\mathrm{U}\left(x\right),\:\:x \in Ω;\xi \in ℝ^{n}.

Nous supposons que a, U > 0 pour x ∈ Ω et que

limxΩU(x)=+.\lim \limits_{x\rightarrow ∂Ω}\mathrm{U}\left(x\right) = + ∞.

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