Retraction maps: A seed of geometric integrators. Part II: Symmetry and reduction
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arXiv
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| Format: | Preprint |
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2025
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| author | Liñán, María Barbero Marrero, Juan Carlos de Diego, David Martín |
| author_facet | Liñán, María Barbero Marrero, Juan Carlos de Diego, David Martín |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_14152 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Retraction maps: A seed of geometric integrators. Part II: Symmetry and reduction Liñán, María Barbero Marrero, Juan Carlos de Diego, David Martín Numerical Analysis Symplectic Geometry 37M15, 65P10, 53D17, 22A22, 70H33 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. |
| title | Retraction maps: A seed of geometric integrators. Part II: Symmetry and reduction |
| topic | Numerical Analysis Symplectic Geometry 37M15, 65P10, 53D17, 22A22, 70H33 |
| url | https://arxiv.org/abs/2502.14152 |