TY - CHAP
T1 - Complete systems of invariants for rank 1 curves in Lagrange Grassmannians
T2 - Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005
Y1 - 2005
A1 - Igor Zelenko
AB - 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.
JF - Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005
UR - http://hdl.handle.net/1963/2310
N1 - Proceedings of 9th Conference on Differential Geometry and its Applications, Prague 2004
U1 - 1706
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -