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

Abstract. This paper studies the construction of geometric integrators for nonholonomic systems. We derive the nonholonomic discrete Euler-Lagrange equations in a setting which permits to deduce geometric integrators for continuous nonholonomic systems (reduced or not). The formalism is given in terms of Lie groupoids, specifying a discrete Lagrangian and a constraint submanifold on it. Additionally, it is necessary to fix a vector subbundle of the Lie algebroid associated to the Lie groupoid. We also discuss the existence of nonholonomic evolution operators in terms of the discrete nonholonomic Legendre transformations and in terms of adequate decompositions of the prolongation of the Lie groupoid. The characterization of the reversibility of the evolution operator and the discrete nonholonomic momentum equation are also considered. Finally, we illustrate with several classical examples the wide range of application of the theory (the discrete nonholonomic constrained particle, the Suslov system, the Chaplygin sleigh, the Veselova system, the rolling ball on a rotating table and the two wheeled planar mobile robot).

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras. Lie bialgebras are examples of generalized Lie bialgebras. Moreover, we prove that the last ones can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structures. We also propose a method to obtain generalized Lie bialgebras. It is a generalization of the YangBaxter equation method. Finally, we describe the structure of a compact generalized Lie bialgebra.Mathematics Subject Classification (2000): 17B62, 22Exx, 53D05, 53D10, 53D17.

Abstract. In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian L d : Q × Q → R, where Q is the configuration manifold, and a (generally nonintegrable) distribution D ⊂ T Q. In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study in particular the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.

We introduce the notion of a symplectic Lie affgebroid and their Lagrangian submanifolds in order to describe the Lagrangian (Hamiltonian) dynamics on a Lie affgebroid in terms of this type of structures. Several examples are discussed.

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangian systems and vakonomic mechanics (variational constrained mechanics). Several examples illustrate the interest of these developments.2000 Mathematics Subject Classification. 17B66, 37J60, 70F25, 70H30, 70H33, 70G45.

In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditions, the reduced space is a poly-Poisson manifold. In addition, some interesting examples of poly-Poisson manifolds are discussed.Under the existence of symmetries, it is possible to perform a reduction procedure in which some of the variables are reduced. The interest of the reduction procedure is twofold: not only we are able to reduce the dynamics but it is a way to generate new examples of symplectic manifolds. One of the reduction methods is the Marsden-Weinstein-Meyer reduction procedure [16]: Given a Hamiltonian action of a Lie group G on a symplectic manifold (M, Ω) with equivariant moment map J : M → g * , it is possible to obtain a symplectic structure on the quotient manifold J −1 (µ)/G µ . An interesting example is the action of a Lie group G on its cotangent bundle T * G by cotangent lifts of left translations and the moment map J : T * G → g * is given by J(α g ) = (T e R g ) * (α g ). Here, the reduced space J −1 (µ)/G µ is just the coadjoint orbit along the element µ ∈ g * endowed with the Kirillov-Kostant-Souriau symplectic structure.On the other hand, the existence of a Lie group of symmetries for a symplectic manifold is one of the justifications for the introduction of Poisson manifolds, which generalize symplectic manifolds. Indeed if (M, Ω) is a symplectic manifold and G is a Lie group acting freely and 2010 Mathematics Subject Classification. 53D05, 53D17, 70645.

Abstract. We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.