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Summary: arXiv:math.DG/0409152
v1
9
Sep
2004
On curvatures and focal points of dynamical Lagrangian
distributions and their reductions by rst integrals
Andrej A. Agrachev Natalia N. Chtcherbakova y Igor Zelenko z
Abstract
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian
distributions, appear naturally in Di erential Geometry, Calculus of Variations and
Rational Mechanics. The basic di erential invariants of a dynamical Lagrangian distri-
bution 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 La-
grangian distribution after its reduction by arbitrary rst integrals in involution. The
interesting phenomenon is that the curvature form of so-called monotone increasing
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