Kowalevski-Sokolov顶的相拓扑
Phase topology of the Kowalevski-Sokolov top
摘要：
研究了$e(3)$上可积Hamilton系统的相拓扑，推广了Kowalevski情形.概括包含，随着一个均匀的势场，陀螺力取决于配置变量。对相对平衡点进行了分类，计算了它们的类型，并定义了稳定性的特征。找到了该情形的Smale图，并给出了两自由度约化系统的等能流形的分类。完整动量映射的临界点集被表示为临界子系统的并集;每个临界子系统都是一个单参数的几乎哈密顿系统族，具有一个自由度。对于所有的临界点，我们明确计算定义其类型的特征值。得到了具有动量映射分岔图的曲面方程。我们给出的例子，现有的等能量图与相应的粗糙拓扑结构（经常刘维尔环面及其分支）的完整描述。
英文摘要：
The phase topology of the integrable Hamiltonian system on $e(3)$ found by V.V.Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations).
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