Juan Carlos Marrero

Lagrangian submanifolds and dynamics on Lie algebroids

In some previous papers, a geometric description of Lagrangian mechanics on Lie algebroids has been developed. In this topical review, we give a Hamiltonian description of mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits us to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange–Poincare (Hamilton–Poincare) equations are the Euler–Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange–Poincare (Hamilton–Poincare) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.

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.

Leibniz algebroid associated with a Nambu-Poisson structure

The notion of a Leibniz algebroid is introduced, and it is shown that each Nambu-Poisson manifold has associated a canonical Leibniz algebroid. This fact permits one to define the modular class of a Nambu-Poisson manifold as an appropriate cohomology class, extending the well known modular class of Poisson manifolds.

Dynamics of generalized Poisson and Nambu–Poisson brackets

A unified setting for generalized Poisson and Nambu–Poisson brackets is discussed. It is proved that a Nambu–Poisson bracket of even order is a generalized Poisson bracket. Characterizations of Poisson morphisms and generalized infinitesimal automorphisms are obtained as coisotropic and Lagrangian submanifolds of product and tangent manifolds, respectively.

Non-holonomic Lagrangian systems in jet manifolds

A geometrical setting in terms of jet manifolds is developed for time-dependent non-holonomic Lagrangian systems. An almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics. A constrained Poincare - Cartan 2-form is defined. If the non-holonomic system is singular, a constraint algorithm is constructed. Special attention is devoted to Caplygin systems and a reduction theorem is proved.

A SURVEY OF LAGRANGIAN MECHANICS AND CONTROL ON LIE ALGEBROIDS AND GROUPOIDS

In this survey, we present a geometric description of Lagrangian and Hamiltonian Mechanics on Lie algebroids. The flexibility of the Lie algebroid formalism allows us to analyze systems subject to nonholonomic constraints, mechanical control systems, Discrete Mechanics and extensions to Classical Field Theory within a single framework. Various examples along the discussion illustrate the soundness of the approach.

Reduction of nonholonomic mechanical systems with symmetries

Abstract A geometric reduction procedure is presented for Lagrangian systems subjected to nonlinear nonholonomic constraints in the presence of symmetries. Our approach is based on a geometrical method which enables one to deduce the constrained dynamics from the unconstrained one by projection.

A geometrical approach to Classical Field Theories: a constraint algorithm for singular theories

We construct a geometrical formulation for first order classical field theories in terms of fibered manifolds and connections. Using this formulation, a constraint algorithm for singular field theories is developed. This algorithm extends the constraint algorithm in mechanics.

Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian mechanics on Lie groupoids. From a variational principle we derive the discrete Euler–Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a symplectic map with respect to the canonical symplectic 2-section on the prolongation of the dual of the Lie algebroid of the given groupoid. The equations we get include the classical discrete Euler–Lagrange equations, the discrete Euler–Poincare and discrete Lagrange–Poincare equations as particular cases. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.

The constraint algorithm for time‐dependent Lagrangians

The aim of this paper is to develop a constraint algorithm for time‐dependent Lagrangian systems which permit us to solve the motion equations. This algorithm extends the Gotay and Nester algorithm for autonomous Lagrangians which is, in fact, a particular case. To do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used.