Manuel de León

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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)
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)
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)
Making use of the geometric setting proposed in [7] to formulate optimal control problems, we address here the treatment of general symmetries. This viewpoint allows us to reduce the number of equations associated with optimal control problems with symmetry and compare the solutions of the original system with the solutions of the reduced one. The(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)