论文标题
低规律性的KAM理论的机械反例
A mechanical counterexample to KAM theory with low regularity
论文作者
论文摘要
我们给出了一个机械示例，即在KAM理论中需要一些规律性。我们考虑了球在定期移动的板上的垂直弹跳运动给出的模型。用$ f $表示板的运动，如果$ \ dot {f} $在norm $ c^5 $中很小，并且每个动作的速度都有限制的速度。如果功能$ f $仅$ c^1 $，则不可能。实际上，我们在c^1 $中构建了一个函数$ f \，其中有任意的小导数在norm $ c^0 $中，其中存在无界速度的运动。
We give a mechanical example concerning the fact that some regularity is necessary in KAM theory. We consider the model given by the vertical bouncing motion of a ball on a periodically moving plate. Denoting with $f$ the motion of the plate, some variants of Moser invariant curve theorem apply if $\dot{f}$ is small in norm $C^5$ and every motion has bounded velocity. This is not possible if the function $f$ is only $C^1$. Indeed we construct a function $f\in C^1$ with arbitrary small derivative in norm $C^0$ for which a motion with unbounded velocity exists.
