非线性哈密顿系统的相对简正模
Relative normal modes for nonlinear Hamiltonian systems
摘要：
给出了具有连续对称性的哈密顿系统在稳定相对平衡点周围的不同相对周期轨道个数的估计。这个结果是对Weinstein和Moser关于稳定平衡点附近能级上周期轨道存在性的经典结果的Hamilton对称框架的推广。所得到的估计是非常精确的，因为它提供了在所讨论的稳定相对平衡点附近的每一个给定能量和动量值下，在任何前缀下，相对周期轨道数目的一个下界（时空）稳定子群。此外，它在特定的例子中是容易计算的。有趣的是，看看如何在我们的结果中，存在的非平凡的相对周期轨道需要（通用）的条件上的高阶项的泰勒展开的哈密顿函数，与纯二次要求的温斯坦-莫泽定理，它强调了高度非线性字符的相对周期的动态对象。
英文摘要：
An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium.The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighboring the stable relative equilibrium in question and with any prefixed (spatiotemporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how in our result the existence of non trivial relative periodic orbits requires (generic) conditions on the higher order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein--Moser Theorem, which emphasizes the highly non linear character of the relatively periodic dynamical objects.
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