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arXiv:math/0409152v1[math.DG]9Sep2004 On curvatures and focal points of dynamical Lagrangian

Summary: arXiv:math/0409152v1[math.DG]9Sep2004
On curvatures and focal points of dynamical Lagrangian
distributions and their reductions by first integrals
Andrej A. Agrachev
Natalia N. Chtcherbakova
Igor Zelenko
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 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 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


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics