Complete systems of invariants for rank 1 curves in Lagrange Grassmannians

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 the results of [4], 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. 1 Statement of the problem and the results Let W be a 2m-dimensional linear space provided with a symplectic form σ. Recall that an m-dimensional subspace Λ of W is called Lagrangian, if σ|Λ = 0. Lagrange Grassmannian L(W ) of W is the set of all Lagrangian subspaces of W . The linear Symplectic group acts naturally on L(W ). Invariants of curves in Lagrange Grassmannian w.r.t. this action are called symplectic The motivation to study differential geometry of curves in Lagrange Grassmannians comes from optimal control problems: it turns out that to any extremal of rather general control systems one can assign a special curve in some Lagrange Grassmannian, called the Jacobi curve (see [1], [2], and Introduction to [4] for the details). Symplectic invariants of Jacobi curves produce curvature-type differential invariants for these control systems. The natural differential-geometric problem is to construct a complete system of symplectic invariants for curves in Lagrange Grassmannians, i.e., some set of invariants such that there exists the unique, up to a symplectic transformation, curve in Lagrange Grassmannian with the prescribed invariants from this set. Some methods for construction and calculation of symplectic invariants of curves in Lagrange Grassmannians (including invariants of unparametrized curves) were given in [4] and [5]. Also the problem of finding a complete system of symplectic invariants for the special class of the so-called rank 1 curves in Lagrange Grassmannians (see Definition 1 ∗S.I.S.S.A., Via Beirut 2-4, 34014, Trieste, Italy; email: [email protected]