David Iglesias

Generalized Lie bialgebroids and Jacobi structures

Abstract 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.

Discrete Nonholonomic Lagrangian Systems on Lie Groupoids

Abstract This paper studies the construction of geometric integrators for nonholonomic systems. We develop a formalism for nonholonomic discrete Euler–Lagrange equations in a setting that 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).

Generalized Lie bialgebras and Jacobi structures on Lie groups

We study generalized Lie bialgebroids over a single point, that is, generalized Lie bialgebras and we prove that they can be considered as the infinitesimal invariants of Lie groups endowed with a certain type of Jacobi structure. We also propose a method generalizing the Yang-Baxter equation method to obtain generalized Lie bialgebras. Finally, we classify the compact generalized Lie bialgebras.

Lie algebroid foliations and ℰ1(M)-Dirac structures

We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid A over M and the leaves of the Lie algebroid foliation on M associated with A. Using these results, we show that a 1(M)-Dirac structure L induces on every leaf F of its characteristic foliation a 1(F)-Dirac structure LF, which comes from a precontact structure or from a locally conformal presymplectic structure on F. In addition, we prove that a Dirac structure on M × can be obtained from L and we discuss the relation between the leaves of the characteristic foliations of L and .

Reduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group

We describe the reduction procedure for a symplectic Lie algebroid by a Lie sub- algebroid 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.

Multisymplectic Geometry and Lie Groupoids

We study higher-degree generalizations of symplectic groupoids, referred to as multisymplectic groupoids. Recalling that Poisson structures may be viewed as infinitesimal counterparts of symplectic groupoids, we describe “higher” versions of Poisson structures by identifying the infinitesimal counterparts of multisymplectic groupoids. Some basic examples and features are discussed.

General framework for nonholonomic mechanics: Nonholonomic systems on Lie affgebroids

This paper presents a geometric description of Lagrangian and Hamiltonian systems on Lie affgebroids subject to affine nonholonomic constraints. We define the notion of nonholonomically constrained system and characterize regularity conditions that guarantee that the dynamics of the system can be obtained as a suitable projection of the unconstrained dynamics. It is shown that one can define an almost aff-Poisson bracket on the constraint AV-bundle, which plays a prominent role in the description of nonholonomic dynamics. Moreover, these developments give a general description of nonholonomic systems and the unified treatment permits to study nonholonomic systems after or before reduction in the same framework. Also, it is not necessary to distinguish between linear or affine constraints and the methods are valid for explicitly time-dependent systems.

Lie algebroid foliations and

We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid A over M and the leaves of the Lie algebroid foliation on M associated with A. Using these results, we show that a 1(M)-Dirac structure L induces on every leaf F of its characteristic foliation a 1(F)-Dirac structure LF, which comes from a precontact structure or from a locally conformal presymplectic structure on F. In addition, we prove that a Dirac structure on M × can be obtained from L and we discuss the relation between the leaves of the characteristic foliations of L and .

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We prove some general results about the relation between the 1-cocycles of an arbitrary Lie algebroid A over M and the leaves of the Lie algebroid foliation on M associated with A. Using these results, we show that a 1(M)-Dirac structure L induces on every leaf F of its characteristic foliation a 1(F)-Dirac structure LF, which comes from a precontact structure or from a locally conformal presymplectic structure on F. In addition, we prove that a Dirac structure on M × can be obtained from L and we discuss the relation between the leaves of the characteristic foliations of L and .