In this paper, it is proved that the infinite KAM torus with prescribed frequency exists in a sufficiently small neighbourhood of a given $I^0$ for nearly integrable and analytic Hamiltonian system $H(I,\theta )=H_0\left(I\right)+ϵH_1(I,\theta )$ of infinite degree of freedom and of short range. That is to say, we will give an extension of the original Kolmogorov theorem to the infinite-dimensional case of short range. The proof is based on the approximation of finite-dimensional Kolmogorov theorem and an improved KAM machinery which works for the normal form depending only on initial $I^0$.

中文翻译：

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关于一些短范围无限维哈密顿系统的Kolmogorov定理

在本文中，证明了给定频率的无限KAM圆环存在于给定的足够小的邻域中 ${\mathrm{一世}}^0$ 用于几乎可积分和解析的哈密顿系统 $H（\mathrm{一世}，\theta ）=H_0（\mathrm{一世}）+ϵH_{\mathrm{1个}}（\mathrm{一世}，\theta ）$具有无限的自由度和短距离。也就是说，我们将原始的Kolmogorov定理扩展到短距离的无穷维情况。该证明是基于有限维Kolmogorov定理的近似值和一种改进的KAM机器，该机器仅以初始形式为标准形式工作${\mathrm{一世}}^0$。