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 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.