Lagrange Grassmannian中秩1曲线的完备不变量系
Complete systems of invariants for rank 1 curves in Lagrange Grassmannians
摘要：
当人们研究与控制系统和几何结构相关的“极值雅可比方程”时，自然会出现拉格朗日格拉斯曼曲线。以这种方式减少了问题的建设的曲率型不变量为这些对象的更具体的问题发现不变量的曲线在拉格朗日格拉斯曼w.r.t.线性辛群的作用。在本文中，我们发展了一种新的方法来微分几何的所谓秩1曲线的拉格朗日格拉斯曼，即，具有速度的曲线是秩为一的线性映射（在拉格朗日格拉斯曼点的切空间与适当的线性映射空间的标准标识下）。这类曲线与“极值的Jacobi方程”相关联，对应于具有标量控制的控制系统和秩2矢量分布。特别是，我们构造的元组的m个主要的不变量，其中m是等于周围的线性辛空间的维数的一半，使得对于一个给定的元组的任意m光滑函数存在唯一的，直到一个辛变换，秩1曲线具有这个元组，作为元组的主要不变量。这种方法推广并简化了我们以前的文章（J. Dynamical and控制系统，8，2002，No.1，93-140）中的一些结果，其中只证明了唯一性部分，而且证明的方式相当繁琐。它是基于构造具有最简单结构方程的新的规范活动标架。
英文摘要：
Curves in Lagrange Grassmannians naturally appear when one studies intrinsically "the Jacobi equations for extremals", associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of finding of invariants of curves in Lagrange Grassmannians w.r.t. the action of the linear Symplectic group. In the present paper we develop a new approach to differential geometry of so-called rank 1 curves in Lagrange Grassmannian, i.e., the curves with velocities being rank one linear mappings (under the standard identification of the tangent space to a point of the Lagrange Grassmannian with an appropriate space of linear mappings). The curves of this class are associated with "the Jacobi equations for extremals", corresponding to control systems with scalar control and to rank 2 vector distributions. In particular, we construct the tuple of m principal invariants, where m is equal to half of dimension of the ambient linear symplectic space, such that for a given tuple of arbitrary m smooth functions there exists the unique, up to a symplectic transformation, rank 1 curve having this tuple, as the tuple of the principal invariants. This approach extends and essentially simplifies some results of our previous paper (J. Dynamical and Control Systems, 8, 2002, No. 1, 93-140), where only the uniqueness part was proved and in rather cumbersome way. It is based on the construction of the new canonical moving frame with the most simple structural equation.
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