This paper gives a review of integration algorithms for finite dimensional
mechanical systems that are based on discrete variational principles. The
variational technique gives a unified treatment of many symplectic schemes,
including those of higher order, as well as a natural treatment of the discrete
Noether theorem. The approach also allows us to include forces, dissipation
and constraints in a natural way. Amongst the many specific schemes treated
as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic
partitioned Runge–Kutta schemes are presented.