一类非经典动力学可积问题的相拓扑
Phase topology of one nonclassical integrable problem of dynamics
摘要：
本文考虑Sokolov和Tsiganov指定Lax表示的三自由度可积系统。Lax表示将A.G.Reyman和M.A.Semenov-Tian-Shansky发现的Kowalevski陀螺仪的L-A对推广到两个恒定场中。本文给出了（几乎处处独立）附加第一积分K和G的显式公式。Sokolov和Tsiganov将这些积分函数与L-A对谱曲线的因子联系起来。由于这种形式的附加积分K和G，没有恒定的陀螺力矩，我们设法找到解析的两个不变的四维子流形上的诱导动力系统几乎处处是两个自由度的哈密顿系统。描述这些不变子流形之一的方程组是刚体动力学中可积Bogoyavlensky情形的不变关系的推广。用临界子系统的方法来描述系统的相拓扑。对于每个子系统，我们构造了分支图，并指定了刘维环面的分支在子系统中，并在系统作为一个整体。
英文摘要：
We consider the integrable system with three degrees of freedom for which Sokolov and Tsiganov specified Lax representation. Lax representation generalizes L-A pair of the Kowalevski gyrostat in two constant fields, found by A.G.Reyman and M.A.Semenov-Tian-Shansky. In the paper, we give the explicit formulas for the (independent almost everywhere) additional first integrals K and G. These integrals are functionally connected with factors of a spectral curve of L-A pair by Sokolov and Tsiganov. Due to this form of additional integrals K and G, without constant gyrostatic moment, we managed to find analytically two invariant four-dimensional submanifolds on which the induced dynamic system is almost everywhere Hamiltonian system with two degrees of freedom. System of equations that describes one of these invariant submanifolds is a generalization of invariant relations of an integrable Bogoyavlensky case in dynamics of a rigid body. To describe phase topology of the system as a whole we use the method of critical subsystems. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of Liouville tori both in subsystems, and in the system as a whole.
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