Generic bifurcation of Hamiltonian vector fields with symmetry
Generic bifurcation of Hamiltonian vector fields with symmetry
复制标题
具有对称性的哈密顿向量场的一般分岔
DOI：
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发表时间：
1992
期刊：
影响因子：
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通讯作者：
J. Marsden
中科院分区：
文献类型：
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作者：
M. Dellnitz;I. Melbourne;J. Marsden
本文的目标之一是明确地描述的一般运动的特征值通过一个一对一的共振线性哈密顿系统，这是关于一个紧李群的辛表示等变。我们对这种运动进行分类，从而回答了这个问题，当碰撞是“危险的”在克莱因的意义上，通过使用组合群论和确定性的相关二次哈密顿量的属性。例如，对于没有对称性或O（2）对称性的系统，一般特征值分裂，而对于具有S1对称性的系统，一般特征值可能分裂或通过。正是在这最后一种情况下，人们必须使用群论和能量学来确定一般的本征值运动。能量学和群论结合的方式总结在表1中。其结果是对比的分歧稳定状态（零特征值），其中可以使用任何一组理论单独（Golubitsky和斯图尔特）或明确的性质的哈密顿量（Cartan-Oh），以确定是否分裂或通过本征值的虚轴。
One of the goals of this paper is to describe explicitly the generic movement of eigenvalues through a one-to-one resonance in a linear Hamiltonian system which is equivariant with respect to a symplectic representation of a compact Lie group. We classify this movement, and hence answer the question of when the collisions are 'dangerous' in the sense of Krein by using a combination of group theory and definiteness properties of the associated quadratic Hamiltonian. For example, for systems with no symmetry or O(2) symmetry generically the eigenvalues split, whereas for systems with S1 symmetry, generically the eigenvalues may split or pass. It is in this last case that one has to use both group theory and energetics to determine the generic eigenvalue movement. The way energetics and group theory are combined is summarized in table 1. The result is to be contrasted with the bifurcation of steady states (zero eigenvalue) where one can use either group theory alone (Golubitsky and Stewart) or definiteness properties of the Hamiltonian (Cartan-Oh) to determine whether the eigenvalues split or pass on the imaginary axis.