Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids
Abstract
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 the classical discrete Euler-Lagrange equations, the discrete Euler-Poincaré and discrete Lagrange-Poincaré equations as particular cases. Our results can be important for the construction of geometric integrators for continuous Lagrangian systems.
- Publication:
-
Nonlinearity
- Pub Date:
- June 2006
- DOI:
- 10.1088/0951-7715/19/6/006
- arXiv:
- arXiv:math/0506299
- Bibcode:
- 2006Nonli..19.1313M
- Keywords:
-
- Mathematics - Differential Geometry;
- Mathematical Physics;
- 17B66;
- 22A22;
- 70G45;
- 70Hxx
- E-Print:
- 38 pages