论文标题
第三和第四步的自由尼尔谎言组的Casimir功能
Casimir functions of free nilpotent Lie groups of steps three and four
论文作者
论文摘要
任何自由的nilpotent Lie代数都由其等级和步骤决定。我们考虑步骤3、4的免费nilpotent Lie代数，并相应地连接并简单地连接的谎言组。 我们构建了此类群体的Casimir函数，即Coadhexhexhight代表的不变性。对于免费的3步尼尔言谎言组，我们可以完整描述Coadexhighine Orbits。事实证明，一般的共同连接轨道是仿射子空间，特殊的共同连接轨道是仿射子空间或非词性四边形的直接产物。 Casimir功能的知识可用于研究动态系统的整合性能和Carnot组的最佳控制问题。特别是，对于三步无卡诺组的某些广泛的时间 - 最佳问题，我们得出结论，与二维的共同连接轨道相对应的极端控制与海森伯格组或恩格尔组的时间优势问题的行为相同。
Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3, 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group.
