  The aim of these notes is to describe how to construct canonical bundles of
moving frames and differential invariants for parametrized curves in Lagrangian
Grassmannians, at least in the monotonic case. Such curves appear as Jacobi
curves of sub-Riemannian extremals [1,2] . Originally this construction was
done in [6,7], where it uses the specifics of Lagrangian Grassmannian. In later
works [3,4] a much more general theory for construction of canonical bundles of
moving frames for parametrized or unparametrized curves in the so-called
generalized flag varieties was developed, so that the problem which is
discussed here can be considered as a particular case of this general theory.
Although this was briefly discussed at the very end of [3], the application of
the theory of [3,4] to obtain the results of [6,7] were never written in detail
and this is our goal here. We believe that this exposition gives a more
conceptual point of view on the original results of [6,7] and especially
clarifies the origin of the normalization conditions of the canonical bundles
of moving frames there, which in fact boil down to a choice of a complement to
a certain subspace of the symplectic Lie algebra. The notes in almost the same
form are included under my authorship as the Appendix in the book "A
Comprehensive Introduction to sub-Riemannian Geometry from Hamiltonian view
point" by A. Agrachev D. Barilari, U. Boscain. Following the style of this
book, some not difficult statements are given as exercises for pedagogical
purposes.
