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Periodic solutions of Lagrangian systems with bounded potential. (English) Zbl 0664.34053
V. Bençi [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 1, 379-400 (1984; Zbl 0561.58006)] studied the existence of T-periodic solutions of the Lagrangian system of ODEs \[ (1)\quad (d/dt)(\partial {\mathcal L}/\partial \xi)(q,\dot q)-(\partial {\mathcal L}/\partial q)(q,\dot q)=0,\quad q\in C^ 2({\mathbb{R}},{\mathbb{R}}^ n) \] where the Lagrangian \[ {\mathcal L}(q,\xi)=()\sum^{n}_{i,j=1}a_{ij}(q)\xi_ i\xi_ j- V(q),\quad q,\xi \in {\mathbb{R}}^ n \] and V(q) is unbounded.
The author studies for bounded V(q) the following problems: (a) The existence of free oscillations of prescribed minimal period T. (b) The existence of multiple free oscillations of period T. (c) The existence of forced oscillations. (d) The case of the double-pendulum.
Reviewer: J.H.Tian

MSC:
34C25 Periodic solutions to ordinary differential equations
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