We present a discrete analog of the recently introduced Hamilton– Pontryagin variational principle in Lagrangian mechanics. This unifies two, previously disparate approaches to discrete Lagrangian mechanics: either using the discrete Lagrangian to define a finite version of Hamilton’s action principle, or treating it as a symplectic generating function. This is demonstrated for a discrete Lagrangian defined on an arbitrary Lie groupoid; the often encountered special case of the pair groupoid (or Cartesian square) is also given as a worked example.
"Discrete Hamilton-Pontryagin mechanics and generating functions on Lie groupoids." J. Symplectic Geom. 8 (2) 225 - 238, June 2010.