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.


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
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