25 research outputs found
Moving constraints as stabilizing controls in classical mechanics
The paper analyzes a Lagrangian system which is controlled by directly
assigning some of the coordinates as functions of time, by means of
frictionless constraints. In a natural system of coordinates, the equations of
motions contain terms which are linear or quadratic w.r.t.time derivatives of
the control functions. After reviewing the basic equations, we explain the
significance of the quadratic terms, related to geodesics orthogonal to a given
foliation. We then study the problem of stabilization of the system to a given
point, by means of oscillating controls. This problem is first reduced to the
weak stability for a related convex-valued differential inclusion, then studied
by Lyapunov functions methods. In the last sections, we illustrate the results
by means of various mechanical examples.Comment: 52 pages, 4 figure
Modified differentials and basic cohomology for Riemannian foliations
We define a new version of the exterior derivative on the basic forms of a
Riemannian foliation to obtain a new form of basic cohomology that satisfies
Poincar\'e duality in the transversally orientable case. We use this twisted
basic cohomology to show relationships between curvature, tautness, and
vanishing of the basic Euler characteristic and basic signature.Comment: 20 pages, references added, minor corrections mad
Energy properness and Sasakian-Einstein metrics
In this paper, we show that the existence of Sasakian-Einstein metrics is
closely related to the properness of corresponding energy functionals. Under
the condition that admitting no nontrivial Hamiltonian holomorphic vector
field, we prove that the existence of Sasakian-Einstein metric implies a
Moser-Trudinger type inequality. At the end of this paper, we also obtain a
Miyaoka-Yau type inequality in Sasakian geometry.Comment: 27 page
Riemannian Gauge Theory and Charge Quantization
In a traditional gauge theory, the matter fields \phi^a and the gauge fields
A^c_\mu are fundamental objects of the theory. The traditional gauge field is
similar to the connection coefficient in the Riemannian geometry covariant
derivative, and the field-strength tensor is similar to the curvature tensor.
In contrast, the connection in Riemannian geometry is derived from the metric
or an embedding space. Guided by the physical principal of increasing symmetry
among the four forces, we propose a different construction. Instead of defining
the transformation properties of a fundamental gauge field, we derive the gauge
theory from an embedding of a gauge fiber F=R^n or F=C^n into a trivial,
embedding vector bundle F=R^N or F=C^N where N>n. Our new action is symmetric
between the gauge theory and the Riemannian geometry. By expressing
gauge-covariant fields in terms of the orthonormal gauge basis vectors, we
recover a traditional, SO(n) or U(n) gauge theory. In contrast, the new theory
has all matter fields on a particular fiber couple with the same coupling
constant. Even the matter fields on a C^1 fiber, which have a U(1) symmetry
group, couple with the same charge of +/- q. The physical origin of this unique
coupling constant is a generalization of the general relativity equivalence
principle. Because our action is independent of the choice of basis, its
natural invariance group is GL(n,R) or GL(n,C). Last, the new action also
requires a small correction to the general-relativity action proportional to
the square of the curvature tensor.Comment: Improved the explanations, added references, added 3 figures and an
appendix, corrected a sign error in the old figure 4 (now figure 5). Now 33
pages, 7 figures and 2 tables. E-mail Serna for annimation
Cohomology of Horizontal Forms
The complex of s-horizontal forms of a smooth foliation F on a manifold M is proved to be exact for every s = 1, . . . , n = codim F, and the cohomology groups of the complex of its global sections, are introduced. They are then compared with other cohomology groups associated to a foliation, previously introduced. An explicit formula for an s-horizontal primitive of an s-horizontal closed form, is given. The problem of representing a de Rham cohomology class by means of a horizontal closed form is analysed. Applications of these cohomology groups are included and several specific examples of explicit computation of such groups-even for non-commutative structure groups-are also presented