Geometric Mechanics and the Dynamics of Asteroid Pairs

This paper studies, using the technique of Lagrangian reduction, the geometric mechanics of a pair of asteroids in orbit about each other under mutual gravitational attraction

Desingularization of Implicit Analytic Differential Equations

The question of finding solutions to given implicit differential equations (IDE) has been answered by several authors in the last few years, using different approaches, in an algebraic and also a geometric setting. Many of those results assume in one way or another that the subimmersion theorem can be applied at several stages of the reduction algorithm, which, roughly speaking, allows to reduce a given IDE to a collection of ODE depending on parameters. The main purpose of the present paper is to improve some of the known results by introducing at each stage of the reduction algorithm a desingularization of the manifolds with singularities that may appear when the subimmersion theorem cannot be applied. This can be done for analytic IDE by using some fundamental results on subanalytic subsets and desingularization of closed subanalytic subsets due mainly to Lojasiewicz, Hironaka, Gabrielov, Hardt, Bierstone, Milman and Sussmann, among others. We will show how this approach helps to understand the dynamics given by the Lagrange-D'Alembert-Poincare equations for the symmetric elastic sphere.Comment: 50 page

The Dirac theory of constraints, the Gotay-Nester theory and Poisson geometry

The Dirac theory of constraints has been widely studied and applied very successfully by physicists since the original works by Dirac and by Bergmann. From a mathematical standpoint, several aspects of the theory have been exposed rigorously afterwards by many authors. However, many questions related to, for instance, singular or infinite dimensional cases remain open. The work of Gotay and Nester presents a mathematical generalization in terms of presymplectic geometry, which introduces a dual point of view. We present a study of the Dirac theory of constraints emphasizing the duality between the Poisson-algebraic and the geometric points of view, related respectively to the work of Dirac and of Gotay and Nester, under strong regularity conditions. We deal with some questions insufficiently treated in the literature: a study of uniqueness of solution; avoiding almost completely the use of coordinates; the role of the Pontryagin bundle. We also show how one can globalize some results usually treated locally in the literature. For instance, we introduce the globalnotion of second class submanifoldas being tangent to a second class subbundle. A general study of global results for Dirac and Gotay-Nester theories remains an open question in this theory.La Teoría de ligaduras deDirac, lateoría de Gotay-Nester y geometría dePoissin. La teoría de Dirac ha sido ampliamente estudiada y aplicada muy exitosamente por los físicos desde los trabajos originales de Dirac y de Bergmann. Desde un punto de vista matemático, varios aspectos de la teoría han sido expuestos rigurosamente por varios autores. Sin embargo, aún quedan abiertas varias preguntas relacionadas, por ejemplo, con casos singulares o infinito-dimensionales. El trabajo de Gotay y Nester presenta una generalización matemática en términos de la geometría presimpléctica, lo cual introduce un punto de vista dual. Presentamos un estudio de la teoría de ligaduras de Dirac enfatizando la dualidad entre los puntos de vista de las álgebras de Poisson y de la geometría presimpléctica, relacionados respectivamente con los trabajos de Dirac y de Gotay-Nester, bajo condiciones de regularidad fuertes. Abordamos algunas cuestiones insuficientemente tratadas en la literatura: un estudio de la unicidad de solución; evitar casi completamente el uso de coordenadas; el rol del fibrado de Pontryagin. También mostramos cómo se pueden globalizar algunos resultados usualmente tratados localmente en la literatura. Por ejemplo, introducimos la noción globalde subvariedad de segunda clasecomo variedad tangente a un subfibrado de segunda clase. Un estudio general de resultados globales para las teorías de Dirac y de Gotay-Nester sigue siendo una pregunta abierta en esta teoría.Fil: Cendra, Hernan. Universidad Nacional del Sur. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca; ArgentinaFil: Etchechoury, María del Rosario. Universidad Nacional de La Plata; ArgentinaFil: Ferraro, Sebastián José. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin

Symmetry Reduction of Optimal Control Systems and Principal Connections

This paper explores the role of symmetries and reduction in nonlinear control and optimal control systems. The focus of the paper is to give a geometric framework of symmetry reduction of optimal control systems as well as to show how to obtain explicit expressions of the reduced system by exploiting the geometry. In particular, we show how to obtain a principal connection to be used in the reduction for various choices of symmetry groups, as opposed to assuming such a principal connection is given or choosing a particular symmetry group to simplify the setting. Our result synthesizes some previous works on symmetry reduction of nonlinear control and optimal control systems. Affine and kinematic optimal control systems are of particular interest: We explicitly work out the details for such systems and also show a few examples of symmetry reduction of kinematic optimal control problems.Comment: 23 pages, 2 figure

A makeshift editorial

Peer Reviewe

Cocycles, compatibility, and Poisson brackets for complex fluids

Motivated by Poisson structures for complex fluids containing cocycles, such as the Poisson structure for spin glasses given by Holm and Kupershmidt in 1988, we investigate a general construction of Poisson brackets with cocycles. Connections with the construction of compatible brackets found in the theory of integrable systems are also briefly discussed

Lagrangian Reduction, the Euler--Poincar\'{e} Equations, and Semidirect Products

There is a well developed and useful theory of Hamiltonian reduction for semidirect products, which applies to examples such as the heavy top, compressible fluids and MHD, which are governed by Lie-Poisson type equations. In this paper we study the Lagrangian analogue of this process and link it with the general theory of Lagrangian reduction; that is the reduction of variational principles. These reduced variational principles are interesting in their own right since they involve constraints on the allowed variations, analogous to what one finds in the theory of nonholonomic systems with the Lagrange d'Alembert principle. In addition, the abstract theorems about circulation, what we call the Kelvin-Noether theorem, are given.Comment: To appear in the AMS Arnold Volume II, LATeX2e 30 pages, no figure