Nonlinear Sciences > Exactly Solvable and Integrable Systems
[Submitted on 20 Apr 2022 (v1), last revised 9 Mar 2023 (this version, v2)]
Title:Lagrangian multiforms on Lie groups and non-commuting flows
View PDFAbstract: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.
Submission history
From: Mats Vermeeren [view email][v1] Wed, 20 Apr 2022 17:55:08 UTC (44 KB)
[v2] Thu, 9 Mar 2023 14:57:54 UTC (46 KB)
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