Caudrelier, V orcid.org/0000-0003-0129-6758, Nijhoff, F, Sleigh, D et al. (1 more author) (2023) Lagrangian multiforms on Lie groups and non-commuting flows. Journal of Geometry and Physics, 187. 104807. ISSN 0393-0440
Abstract
We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as a variational description of integrable systems in the sense of multidimensional consistency. In the context of non-commuting flows, the manifold of independent variables, often called multi-time, is a Lie group whose bracket structure corresponds to the commutation relations between the vector fields generating the flows. Natural examples are provided by superintegrable systems for the case of Lagrangian 1-form structures, and integrable hierarchies on loop groups in the case of Lagrangian 2-forms. As particular examples we discuss the Kepler problem, the rational Calogero-Moser system, and a generalisation of the Ablowitz-Kaup-Newell-Segur system with non-commuting flows. We view this endeavour as a first step towards a purely variational approach to Lie group actions on manifolds.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © 2023 The Author(s). This is an open access article under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited. |
Keywords: | Integrable hierarchies, Non-commuting flows, Lagrangian multiforms, Symmetry algebras |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 20 Mar 2023 14:57 |
Last Modified: | 25 Jun 2023 23:16 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.geomphys.2023.104807 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:196915 |