Retraction maps: A seed of geometric integrators. Part II: Symmetry and reduction

Abstract

In this paper we use retraction and discretization maps (see [Barbero Li\~n\'an and Mart\'in de Diego, 2022]) as a tool for deriving in a systematic way numerical integrators preserving geometric structures (such as symplecticity or Lie-Poisson structure), as well as methods that preserve symmetry and the associated discrete momentum map. The classical notion of a retraction map leads to the notion of discretization map extended here to the Lie algebroid of a Lie groupoid so that the configuration manifold is discretized, instead of the equations of motion. As a consequence, geometric integrators are obtained preserving the Lie-Poisson structure of the corresponding reduced system.