A new perspective on symplectic integration of constrained mechanical systems via discretization maps

María Barbero Liñán; David Martín de Diego; Rodrigo T. Sato Martín de Almagro

doi:10.3934/jcd.2025009

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2026, Volume 13: 1-28. Doi: 10.3934/jcd.2025009

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A new perspective on symplectic integration of constrained mechanical systems via discretization maps

1.
Departamento de Matemíatica Aplicada, Universidad Politíecnica de Madrid, Av. Juan de Herrera 4, 28040 Madrid, Spain
2.
Instituto de Ciencias Matemíaticas (CSIC-UAM-UC3M-UCM), C/Nicolías Cabrera 13-15, 28049 Madrid, Spain
3.
Friedrich-Alexander-Universität Erlangen-Nürnberg, Institute of Applied Dynamics, Immerwahrstrasse 1, 91058 Erlangen, Germany

^*Corresponding author: María Barbero Liñán
^*Corresponding author: María Barbero Liñán

Received: November 2024

Revised: August 2025

Published online: November 9, 2025

The authors are supported by PID2022-137909NB-C21, RED2022-134301-TD, PCI2024-155047-2 and CEX2023-001347-S funded by MCIN/AEI/10.13039/501100011033.

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Abstract

A new geometric procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the discretization rule rather than vice versa. As a result, the constraint submanifold is exactly preserved by the symplectic discrete flow and the extension of these methods to the case of non-linear configuration spaces, such as systems on Lie groups with holonomic constraints, is also described.

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Mathematics Subject Classification: Primary: 65P10, 70G45, 70G65; Secondary: 37M15, 53D12, 53D22.

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References

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