In 1978, Rabinowitz proved the existence of a non-constant T-periodic solution for nonlinear Hamiltonian systems on R2n with Hamiltonian function being super-quadratic at the infinity and zero for any given T > 0. Since the minimal period of this solution may be T/k for some positive integer k, he proposed the question whether there exists a solution with T as its minimal period for such a Hamiltonian system. This is the so-called Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, this conjecture has been deeply studied by many mathematicians. But under the original structural conditions of Rabinowitz, the conjecture is still open when n ≥ 2. In this paper, I give a brief survey on the studies of this conjecture and hope to lead to more interests on it.

中文翻译：

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拉比诺维茨最小周期解猜想

1978 年，Rabinowitz 证明了非常数的存在吨-非线性哈密顿系统的周期解R2n哈密​​顿函数在无穷大处是超二次的，对于任何给定的都为零吨 > 0. 由于此解决方案的最短周期可能是吨/ķ对于一些正整数ķ，他提出了是否存在解决方案的问题吨作为这种哈密顿系统的最小周期。这就是所谓的拉比诺维茨最小周期解猜想. 在过去的 40 多年里，这一猜想得到了许多数学家的深入研究。但在拉比诺维茨原来的结构条件下，当n ≥ 2. 在本文中，我对这个猜想的研究进行了简要的概述，希望能引起更多的兴趣。