Acta Numerica 2001

Discrete mechanics and variational integrators

J. E. Marsden a1 and M. West a1
a1 Control and Dynamical Systems 107-81, Caltech, Pasadena, CA 91125-8100, USA E-mail:,


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.