KAM theorem of symplectic algorithms for Hamiltonian systems
KAM theorem of symplectic algorithms for Hamiltonian systems
复制标题
DOI：
10.1007/s002110050460
复制
发表时间：
1999-09
影响因子：
2.1
通讯作者：
Zaijiu Shang
中科院分区：
文献类型：
--
作者：
Zaijiu Shang
在本文中，我们证明了著名的KAM定理的一个类似物对辛算法成立，这是Channel和Scovel（1990），Feng Kang（1991）和Sanz-Serna和Calvo（1994）几年前提出的。主要结果包括：当辛数值算法应用于一般可积Hamilton系统时，如果系统是解析的，且算法的时间步长足够小，则存在具有光滑叶理结构的不变环面.这个存在性结果也意味着，当它被应用到一个一般的可积系统，具有nindependent光滑不变函数，这些函数是在对合和良好定义的集合上的不变环面在Whitney意义下填充。不变环面就是这些函数的水平集。给出了算法的数值不变环面逼近系统的精确不变环面的一些定量结果。
In this paper we prove that an analog of the celebrated KAM theorem holds for symplectic algorithms, which Channel and Scovel (1990), Feng Kang (1991) and Sanz-Serna and Calvo (1994) suggested a few years ago. The main results consist of the existence of invariant tori, with a smooth foliation structure, of a symplectic numerical algorithm when it applies to a generic integrable Hamiltonian system if the system is analytic and the time-step size of the algorithm is s ufficiently small. This existence result also implies that the algorithm, when it is applied to a generic integrable system, possessesnindependent smooth invariant functions which are in involution and well-defined on the set filled by the invariant tori in the sense of Whitney. The invariant tori are just the level sets of these functions. Some quantitative results about the numerical invariant tori of the algorithm approximating the exact ones of the system are also given.