Retraction maps: A seed of geometric integrators. Part II: Symmetry and reduction
María Barbero Liñán, Juan Carlos Marrero, David Martín de Diego
TL;DR
This work develops a unified framework for structure-preserving numerical integration by extending retraction and discretization maps from classical manifolds to Lie groupoids and their Lie algebroids. By incorporating tangent and cotangent lifts and reduction under symmetries, the authors derive Lie-Poisson and Poisson integrators for reduced systems, including Lie-Poisson integrators on Lie groups and action Lie algebroids. They present explicit constructions for linear and higher-order integrators, using compositions of Lagrangian submanifolds to achieve elevated order while preserving Poisson structures. The framework promises broad applicability to mechanical systems with symmetry, such as rigid body dynamics and heavy tops, and suggests potential applications in optimization and neural networks with symmetry.
Abstract
In this paper we use retraction and discretization maps (see [Barbero Liñán and Martín 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.