Lagrangian submanifolds and dynamics on Lie algebroids

@article{Len2004LagrangianSA,
  title={Lagrangian submanifolds and dynamics on Lie algebroids},
  author={Manuel de Le{\'o}n and Juan Carlos Marrero and Eduardo Mart{\'i}nez},
  journal={Journal of Physics A},
  year={2004},
  volume={38}
}

Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids

The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified

Lagrangian Lie Subalgebroids Generating Dynamics for Second-Order Mechanical Systems on Lie Algebroids

The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified

Classical field theory on Lie algebroids: Multisymplectic formalism

The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric

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

Hamiltonian Mechanics on Duals of Generalized Lie Algebroids

A new description, different by the classical theory of Hamiltonian Mechanics, in the general framework of generalized Lie algebroids is presented. In the particular case of Lie algebroids, new and

Calculus on Lie algebroids, Lie groupoids and Poisson manifolds

We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a

Dynamical equations and Lagrange–Ricci flow evolution on prolongation Lie algebroids

The approach to nonholonomic Ricci flows and geometric evolution of regular Lagrange systems (S. Vacaru. J. Math. Phys. 49, 043504 (2008); Ibid. Rep. Math. Phys. 63, 95 (2009)) is extended to include

Legendre Duality Between Lagrangian and Hamiltonian Mechanics

In some previous papers, a Legendre duality between Lagrangian and Hamiltonian Mechanics has been developed. The (\rho,\eta)-tangent application of the Legendre bundle morphism associated to a

Dirac cotangent bundle reduction

The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie--Dirac reduction . This procedure simultaneously
...

References

SHOWING 1-10 OF 44 REFERENCES

Classical field theory on Lie algebroids: Multisymplectic formalism

The jet formalism for Classical Field theories is extended to the setting of Lie algebroids. We define the analog of the concept of jet of a section of a bundle and we study some of the geometric

Variational principles for Lie-Poisson and Hamilton-Poincaré equations

As is well-known, there is a variational principle for the Euler–Poincare equations on a Lie algebra g of a Lie group G obtained by reducing Hamilton’s principle on G by the action of G by, say,

Lie algebroid structures on a class of affine bundles

We introduce the notion of a Lie algebroid structure on an affine bundle whose base manifold is fibered over R. It is argued that this is the framework which one needs for coming to a time-dependent

Lagrangian Reduction by Stages

This booklet studies the geometry of the reduction of Lagrangian systems with symmetry in a way that allows the reduction process to be repeated; that is, it develops a context for Lagrangian

Classical field theory on Lie algebroids: variational aspects

The variational formalism for classical field theories is extended to the setting of Lie algebroids. Given a Lagrangian function, we study the problem of finding critical points of the action

Lie algebroid structures and Lagrangian systems on affine bundles

Lagrangian Mechanics on Lie Algebroids

A geometric description of Lagrangian Mechanics on Lie algebroids is developed in a parallel way to the usual formalism of Lagrangian Mechanics on the tangent bundle of a manifold. The dynamical

AV-differential geometry: Poisson and Jacobi structures☆

Lie bialgebroids and Poisson groupoids

Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a

Deformation Quantization of Poisson Manifolds

I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the