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- Manuel de León, Juan C. Marrero, Eduardo Mart́ınez
- 2004

In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, 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… (More)

- M. DE LEÓN
- 2002

A new geometrical setting for classical field theories is introduced. This description is strongly inspired in the one due to Skinner and Rusk for singular lagrangians systems. For a singular field theory a constraint algorithm is developed that gives a final constraint submanifold where a well-defined dynamics exists. The main advantage of this algorithm… (More)

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… (More)

This paper presents a geometric description on Lie algebroids of Lagrangian systems subject to nonholonomic constraints. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of nonholonomically constrained system, and characterize regularity conditions that guarantee the dynamics of the… (More)

- J. Cortés, Manuel de León, David Martín de Diego, S. Martínez
- SIAM J. Control and Optimization
- 2002

We treat the vakonomic dynamics with general constraints within a new geometric framework which can be useful in the study of optimal control problems. We compare our formulation with the one of Vershik and Gershkovich in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description which allows to develop an… (More)

The reduction and reconstruction of the dynamics of nonholonomic mechanical systems with symmetry are investigated. We have considered a more general framework of constrained hamiltonian systems since they appear in the reduction procedure. A reduction scheme in terms of the nonholonomic momentum mapping is developed. The reduction of the nonholonomic… (More)

There are several numerical integration methods [16] that preserve some of the invariants of an autonomous mechanical system. In [8], T.D. Lee studies the possibility that time can be regarded as a bona fide dynamical variable giving a discrete time formulation of mechanics (see also [9, 10]). From other point of view (integrability aspects) Veselov [19]… (More)

In this paper we extend the geometric formalism of the Hamilton-Jacobi theory for hamiltonian mechanics to the case of classical field theories in the framework of multisymplectic geometry and Ehresmann connections.

In this paper, we study the underlying geometry in the classical Hamilton-Jacobi theory. The proposed formalism is also valid for nonholonomic systems. We first introduce the essential geometric ingredients: a vector bundle, a linear almost Poisson structure and a Hamiltonian function, both on the dual bundle (a Hamiltonian system). From them, it is… (More)

The constraint distribution in non-holonomic mechanics has a double role. On one hand, it is a kinematic constraint, that is, it is a restriction on the motion itself. On the other hand, it is also a restriction on the allowed variations when using D’Alembert’s Principle to derive the equations of motion. We will show that many systems of physical interest… (More)