In this article, we study the existence of periodic solutions to second order Hamiltonian systems. Our goal is twofold. When the nonlinear term satisfies a strictly monotone condition, we show that, for any $T>0$ , there exists a T-periodic solution with minimal period T. When the nonlinear term satisfies a non-decreasing condition, using a perturbation technique, we prove a similar result. In the latter case, the periodic solution corresponds to a critical point which minimizes the variational functional on the Nehari manifold which is not homeomorphic to the unit sphere.

中文翻译：

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具有哈密顿系统规定最小周期的周期解

在本文中，我们研究了二阶哈密顿系统的周期解的存在性。我们的目标是双重的。当非线性项满足严格的单调条件时，我们表明，对于任何$ T> 0 $，都存在具有最小周期T的T周期解。当非线性项满足非递减条件时，使用摄动技术，我们证明了类似的结果。在后一种情况下，周期解对应于一个临界点，该临界点使Nehari流形上的变函数最小化，该函数对单位球体不是同胚的。