The reduced spaces of a symplectic Lie group action

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino. In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.Comment: 37 page

The optimal momentum map

The presence of symmetries in a Hamiltonian system usually implies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden--Weinstein reduction procedure takes advantage of this and associates to the original system a new Hamiltonian system with fewer degrees of freedom. However, in a large number of situations, this standard approach does not work or is not efficient enough, in the sense that it does not use all the information encoded in the symmetry of the system. In this work, a new momentum map will be defined that is capable of overcoming most of the problems encountered in the traditional approach.Comment: 35 pages. To appear in: Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J.E. Marsden. P. Holmes, P. Newton, and A. Weinstein, eds., Springer-Verlag, New York, 200

A symplectic slice theorem

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model it can be used the so called Chu map instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.Comment: 14 page

Unveiling the circumstellar environment towards a massive young stellar object

As a continuation of a previous work, in which we found strong evidence of massive molecular outflows towards a massive star forming site, we present a new study of this region based on very high angular resolution observations with the aim of discovering the outflow driven mechanism. Using near-IR data acquired with Gemini-NIRI at the broad H- and Ks-bands, we study a region of 22" x 22" around the UCHII region G045.47+0.05, a massive star forming site at the distance of about 8 kpc. To image the source with the highest spatial resolution possible we employed the adaptative optic system ALTAIR, achieving an angular resolution of about 0.15". We discovered a cone-like shape nebula with an opening angle of about 90 degree extending eastwards the IR source 2MASS J19142564+1109283, a very likely MYSO. This morphology suggests a cavity that was cleared in the circumstellar material and its emission may arise from scattered continuum light, warm dust, and likely emission lines from shock-excited gas. The nebula, presenting arc-like features, is connected with the IR source through a jet-like structure, which is aligned with the blue shifted CO outflow found in a previous study. The near-IR structure lies ~3" north of the radio continuum emission, revealing that it is not spatially coincident with the UCHII region. The observed morphology and structure of the near-IR nebula strongly suggest the presence of a precessing jet. In this study we have resolved the circumstellar ambient (in scale of a thousand A.U.) of a distant MYSO, indeed one of the farthest cases.Comment: Accepted in A&A Letters (October 2013

Energy Shaping Control of an Inverted Flexible Pendulum Fixed to a Cart

Control of compliant mechanical systems is increasingly being researched for several applications including flexible link robots and ultra-precision positioning systems. The control problem in these systems is challenging, especially with gravity coupling and large deformations, because of inherent underactuation and the combination of lumped and distributed parameters of a nonlinear system. In this paper we consider an ultra-flexible inverted pendulum on a cart and propose a new nonlinear energy shaping controller to keep the pendulum at the upward position with the cart stopped at a desired location. The design is based on a model, obtained via the constrained Lagrange formulation, which previously has been validated experimentally. The controller design consists of a partial feedback linearization step followed by a standard PID controller acting on two passive outputs. Boundedness of all signals and (local) asymptotic stability of the desired equilibrium is theoretically established. Simulations and experimental evidence assess the performance of the proposed controller.Comment: 11 pages, 7 figures, extended version of the NOLCOS 2016 pape

The Mexican Animal Identification System: Current Situation, Problems, and Potential

Mexico initiated a federal animal identification (ID) system (SINIIGA) in 2003. The program is administered by an agency of the federal Department of Agriculture (SAGARPA) and has been used primarily to support a federal subsidy program for livestock producers. The program is conceptually well designed, but implementation thus far falls short of the potential and needs, most importantly in animal disease management. Although substantial numbers of animals have been tagged, relatively little progress has been made in developing a usable animal ID information system. Animal health officials currently are not actively involved in the development and use of the system.Mexican animal ID system, livestock, Mexico, Agribusiness, Agricultural and Food Policy, Farm Management, International Development, Livestock Production/Industries, Political Economy, Q13, Q18,