Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids
Marrero, J. C. ; de Diego, D. Martín ; Martínez, E.
arXiv, 0506299 / Harvested from arXiv
Text on arXiv
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 as particular cases the classical discrete Euler-Lagrange equations, the discrete Euler-Poincar\'e and discrete Lagrange-Poincar\'e equations. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.
Publié le : 2005-06-15
Classification:  Mathematics - Differential Geometry,  Mathematical Physics,  17B66,  22A22,  70G45,  70Hxx 
@article{0506299,
     author = {Marrero, J. C. and de Diego, D. Mart\'\i n and Mart\'\i nez, E.},
     title = {Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0506299}
}

Marrero, J. C.; de Diego, D. Martín; Martínez, E. Discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0506299/