Discrete geometric mechanics for variational time integrators

In this chapter, we present a geometric--instead of a traditional numerical-analytic--approach to the problem of time integration. Geometry at its most abstract is the study of symmetries and their associated invariants. Variational approaches based on such notions are commonly used in geometric modeling and discrete differential geometry. Here we will treat mechanics in a similar way. Indeed, the very essence of a mechanical system is characterized by its symmetries and invariants. Thus preserving these symmetries and invariants (e.g., certain momenta) into the discrete computational setting is of paramount importance if one wants discrete time integration to properly capture the underlying continuous motion. Motivated by the well-known variational and geometric nature of most dynamical systems, we review the use of discrete variational principles as a way to derive robust, and accurate time integrators.