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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 alge-broids 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 sub-manifold where a well-defined dynamics exists. The main advantage of this algorithm… (More)

In this paper, we study the underlying geometry in the classical Hamilton-Jacobi equation. 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)

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)

This review paper is concerned with the generalizations to field theory of the tangent and cotangent structures and bundles that play fundamental roles in the Lagrangian and Hamiltonian formulations of classical mechanics. The paper reviews, compares and constrasts the various generalizations in order to bring some unity to the field of study. The… (More)

6 The case of horizontal symmetries 29 6. Abstract 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… (More)

We discuss an extension of the Hamilton–Jacobi theory to non-holonomic mechanics with a particular interest in its application to exactly integrating the equations of motion. We give an intrinsic proof of a nonholo-nomic analogue of the Hamilton–Jacobi theorem. Our intrinsic proof clarifies the difference from the conventional Hamilton–Jacobi theory for… (More)

A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the " gyrolanguage " of this paper one attaches the prefix " gyro " to a classical term to mean the… (More)

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)

- Manuel de León, Aitor Santamaŕıa-Merino
- 2003

In this paper the notion of Tulczyjew's triples in classical mechanics is extended to classical field theories, using the so-called multisymplectic formalism, and a convenient notion of lagrangian submanifold in multisymplectic geometry. Accordingly, the dynamical equations are interpreted as the local equations defining these la-grangian submanifolds.