Lagrangian multiforms on Lie groups and non-commuting flows

TL;DR

This paper develops a variational framework for non-commuting flows on Lie groups, extending Lagrangian multiform theory to describe integrable systems with non-abelian symmetries and providing concrete examples like the Kepler problem.

Contribution

It introduces a novel variational approach for non-commuting flows on Lie groups, generalizing Lagrangian multiforms to non-abelian settings and illustrating with key integrable systems.

Findings

Extended Lagrangian multiform theory to Lie group actions.

Applied framework to classical integrable systems like Kepler and Calogero-Moser.

Provided examples of non-commuting flows in integrable hierarchies.

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.