Generic Hamiltonian dynamical systems are neither integrable nor ergodic
Generic Hamiltonian dynamical systems are neither integrable nor ergodic
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通用哈密顿动力系统既不可积也不可遍历
DOI：
10.1090/memo/0144
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发表时间：
1974
影响因子：
1.9
通讯作者：
K. Meyer
中科院分区：
文献类型：
--
作者：
L. Markus;K. Meyer
本文考虑2n维紧致辛流形M上的光滑Hamilton微分方程。所有这样的方程的空间被认为是一个完整的度量空间，通过放置通常的C拓扑上的归一化哈密顿。证明了所有允许n个独立对合积分的Hamilton方程的集合是第一Baire范畴。同时证明了在稠密能级集上遍历的所有哈密顿方程的集合也是第一Baire范畴。因此，在一般情况下，哈密顿方程既不是遍历的，也不是可积的。
This paper considers smooth Hamiltonian differential equations on a compact symplectic manifold M of dimension 2n. The space of all such equations is considered as a complete metric space by placing the usual C topology on the normalized Hamiltonians. It is shown that the set of all Hamiltonians equations which admit n independent integrals in involution is of first Baire category. At the same time it is shown that the set of all Hamiltonian equations which are ergodic on a dense set of energy levels is also of first Baire category. Thus Hamiltonian equations are neither ergodic nor integrable in the generic case.