  Curves in Lagrange Grassmannians appear naturally in the intrinsic study of
geometric structures on manifolds. By a smooth geometric structure on a
manifold we mean any submanifold of its tangent bundle, transversal to the
fibers. One can consider the time-optimal problem naturally associate with a
geometric structure. The Pontryagin extremals of this optimal problem are
integral curves of certain Hamiltonian system in the cotangent bundle. The
dynamics of the fibers of the cotangent bundle w.r.t. this system along an
extremal is described by certain curve in a Lagrange Grassmannian, called
Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves
produces the invariant of the original geometric structure. The basic
characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The
number of boxes in its kth column is equal to the rank of the kth derivative of
the curve (which is an appropriately defined linear mapping) at a generic
point. We will describe the construction of the complete system of symplectic
invariants for parameterized curves in a Lagrange Grassmannian with given Young
diagram. It allows to develop in a unified way local differential geometry of
very wide classes of geometric structures on manifolds, including both
classical geometric structures such as Riemannian and Finslerian structures and
less classical ones such as sub-Riemannian and sub-Finslerian structures,
defined on nonholonomic distributions.
