  Pairs (Hamiltonian system, Lagrangian distribution), called dynamical
Lagrangian distributions, appear naturally in Differential Geometry, Calculus
of Variations and Rational Mechanics. The basic differential invariants of a
dynamical Lagrangian distribution w.r.t. the action of the group of
symplectomorphisms of the ambient symplectic manifold are the curvature
operator and the curvature form. These invariants can be seen as
generalizations of the classical curvature tensor in Riemannian Geometry. In
particular, in terms of these invariants one can localize the focal points
along extremals of the corresponding variational problems. In the present paper
we study the behavior of the curvature operator, the curvature form and the
focal points of a dynamical Lagrangian distribution after its reduction by
arbitrary first integrals in involution. The interesting phenomenon is that the
curvature form of so-called monotone increasing Lagrangian dynamical
distributions, which appear naturally in mechanical systems, does not decrease
after reduction. It also turns out that the set of focal points to the given
point w.r.t. the monotone increasing dynamical Lagrangian distribution and the
corresponding set of focal points w.r.t. its reduction by one integral are
alternating sets on the corresponding integral curve of the Hamiltonian system
of the considered dynamical distributions. Moreover, the first focal point
corresponding to the reduced Lagrangian distribution comes before any focal
point related to the original dynamical distribution. We illustrate our results
on the classical $N$-body problem.
