## Normal modes of a Lagrangian system constrained in a potential well.(English)Zbl 0561.58006

Let a, $$U\in C^ 2(\Omega)$$ where $$\Omega$$ is a bounded set in $${\mathbb{R}}^ n$$ and let (*) $$L(x,\xi)=a(x)| \xi |^ 2-U(x)$$, $$x\in \Omega$$; $$\xi \in {\mathbb{R}}^ n$$. We suppose that a, $$U>0$$ for $$x\in \Omega$$ and that $$\lim_{x\to \partial \Omega}U(x)=+\infty$$. 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.

### MSC:

 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 34C25 Periodic solutions to ordinary differential equations 70K99 Nonlinear dynamics in mechanics

### Keywords:

Lagrangian system; periodic solutions
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### References:

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