Some linear Jacobi structures on vector bundles

We study Jacobi structures on the dual bundle $A^\ast$ to a vector bundle $A$ such that the Jacobi bracket of linear functions is again linear and the Jacobi bracket of a linear function and the constant function 1 is a basic function. We prove that a Lie algebroid structure on $A$ and a 1-cocycle $\phi \in \Gamma (A^\ast)$ induce a Jacobi structure on $A^\ast$ satisfying the above conditions. Moreover, we show that this correspondence is a bijection. Finally, we discuss some examples and applications.Comment: 6 pages, To appear in C. R. Acad. Sci. Paris, S\'erie

Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane

It is known that the reduced equations for an axially symmetric homogeneous ellipsoid that rolls without slipping on the plane possess a smooth invariant measure. We show that such an invariant measure does not exist in the case when all of the semi-axes of the ellipsoid have different length.Comment: v2: Minor changes after journal review. This text uses the theory developed in arXiv:1304.1788 for the specific example of a homogeneous ellipsoid rolling on the plan

Unimodularity and preservation of volumes in nonholonomic mechanics

The equations of motion of a mechanical system subjected to nonholonomic linear constraints can be formulated in terms of a linear almost Poisson structure in a vector bundle. We study the existence of invariant measures for the system in terms of the unimodularity of this structure. In the presence of symmetries, our approach allows us to give necessary and sufficient conditions for the existence of an invariant volume, that unify and improve results existing in the literature. We present an algorithm to study the existence of a smooth invariant volume for nonholonomic mechanical systems with symmetry and we apply it to several concrete mechanical examples.Comment: 37 pages, 3 figures; v3 includes several changes to v2 that were done in accordance to the referee suggestion

The inhomogeneous Suslov problem

We consider the Suslov problem of nonholonomic rigid body motion with inhomogeneous constraints. We show that if the direction along which the Suslov constraint is enforced is perpendicular to a principal axis of inertia of the body, then the reduced equations are integrable and, in the generic case, possess a smooth invariant measure. Interestingly, in this generic case, the first integral that permits integration is transcendental and the density of the invariant measure depends on the angular velocities. We also study the Painlev\'e property of the solutions.Comment: 10 pages, 5 figure

Poisson-Lie groups, bi-Hamiltonian systems and integrable deformations

Producción CientíficaGiven a Lie-Poisson completely integrable bi-Hamiltonian system on R^n, we present a method which allows us to construct, under certain conditions, a completely integrable bi-Hamiltonian deformation of the initial Lie-Poisson system on a non-abelian Poisson-Lie group G_eta of dimension n, where eta \in R is the deformation parameter. Moreover, we show that from the two multiplicative (Poisson-Lie) Hamiltonian structures on G_eta that underly the dynamics of the deformed system and by making use of the group law on G_eta, one may obtain two completely integrable Hamiltonian systems on G_eta x G_eta. By construction, both systems admit reduction, via the multiplication in G_eta, to the deformed bi-Hamiltonian system in G_eta. The previous approach is applied to two relevant Lie-Poisson completely integrable bi-Hamiltonian systems: the Lorenz and Euler top systems

A new canonical affine BRACKET formulation of Hamiltonian Classical Field theories of first order

It has been a long standing question how to extend the canonical Poisson bracket formulation from classical mechanics to classical field theories, in a completely general, intrinsic, and canonical way. In this paper, we provide an answer to this question by presenting a new completely canonical bracket formulation of Hamiltonian Classical Field Theories of first order on an arbitrary configuration bundle. It is obtained via the construction of the appropriate field-theoretic analogues of the Hamiltonian vector field and of the space of observables, via the introduction of a suitable canonical Lie algebra structure on the space of currents (the observables in field theories). This Lie algebra structure is shown to have a representation on the affine space of Hamiltonian sections, which yields an affine analogue to the Jacobi identity for our bracket. The construction is analogous to the canonical Poisson formulation of Hamiltonian systems although the nature of our formulation is linear-affine and not bilinear as the standard Poisson bracket. This is consistent with the fact that the space of currents and Hamiltonian sections are respectively, linear and affine. Our setting is illustrated with some examples including Continuum Mechanics and Yang-Mills theory

Generalized Lie bialgebroids and Jacobi structures

The notion of a generalized Lie bialgebroid (a generalization of the notion of a Lie bialgebroid) is introduced in such a way that a Jacobi manifold has associated a canonical generalized Lie bialgebroid. As a kind of converse, we prove that a Jacobi structure can be defined on the base space of a generalized Lie bialgebroid. We also show that it is possible to construct a Lie bialgebroid from a generalized Lie bialgebroid and, as a consequence, we deduce a duality theorem. Finally, some special classes of generalized Lie bialgebroids are considered: triangular generalized Lie bialgebroids and generalized Lie bialgebras.Comment: 32 page

Functional Lateralization of Temporoparietal Junction: Imitation Inhibition, Visual Perspective Taking and Theory of Mind.

Although neuroimaging studies have consistently identified the temporoparietal junction (TPJ) as a key brain region involved in social cognition, the literature is far from consistent with respect to lateralization of function. For example, bilateral TPJ activation is found during theory of mind tasks in some studies, but only right hemisphere activation in others. Visual perspective taking and imitation inhibition, which have been argued to recruit the same socio-cognitive processes as theory of mind, are associated with unilateral activation of either left TPJ (perspective taking), or right TPJ (imitation inhibition). The present study investigated the functional lateralization of TPJ involvement in the above three socio-cognitive abilities using transcranial direct current stimulation. Three groups of healthy adults received anodal stimulation over right TPJ, left TPJ or the occipital cortex prior to performing three tasks (imitation inhibition, visual perspective taking and theory of mind). In contrast to the extant neuroimaging literature, our results suggest bilateral TPJ involvement in imitation inhibition and visual perspective taking, while no effect of anodal stimulation was observed on theory of mind. The discrepancy between these findings and those obtained using neuroimaging highlight the efficacy of neurostimulation as a complementary methodological tool in cognitive neuroscience. This article is protected by copyright. All rights reserved