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

María Barbero Liñán; Juan Carlos Marrero; David Martín de Diego

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arXiv · 2502.14152

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

María Barbero Liñán·Juan Carlos Marrero·David Martín de Diego

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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.

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María Barbero Liñán, Juan Carlos Marrero, David Martín de Diego. 2025-02-19. Retraction maps: A seed of geometric integrators. Part II: Symmetry and reduction. https://arxiv.org/abs/2502.14152

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