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Lagrangian multiforms on Lie groups and non-commuting flows

Vincent Caudrelier, Frank Nijhoff, Duncan Sleigh, Mats Vermeeren

TL;DR

This work extends the variational framework of Lagrangian multiforms to noncommuting flows by making the multi-time a Lie group, with the group structure encoding the commutation relations of the flows. It develops jet bundles on Lie groups, formulates a variational principle for functions on the group, and constructs Lagrangian 1-forms and 2-forms that capture noncommuting dynamics, including a nonlinear Poisson setting and a noncommutative AKNS hierarchy. The approach yields concrete results for superintegrable systems such as Kepler and Calogero-Moser, and provides SE(2) and infinite hierarchy examples that illustrate noncommutative zero-curvature relations and symmetry-generated evolutions. Collectively, the paper offers a first-principles variational description of Lie group actions on manifolds and paves the way for purely variational treatments of noncommuting symmetry flows and their associated 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.

Lagrangian multiforms on Lie groups and non-commuting flows

TL;DR

This work extends the variational framework of Lagrangian multiforms to noncommuting flows by making the multi-time a Lie group, with the group structure encoding the commutation relations of the flows. It develops jet bundles on Lie groups, formulates a variational principle for functions on the group, and constructs Lagrangian 1-forms and 2-forms that capture noncommuting dynamics, including a nonlinear Poisson setting and a noncommutative AKNS hierarchy. The approach yields concrete results for superintegrable systems such as Kepler and Calogero-Moser, and provides SE(2) and infinite hierarchy examples that illustrate noncommutative zero-curvature relations and symmetry-generated evolutions. Collectively, the paper offers a first-principles variational description of Lie group actions on manifolds and paves the way for purely variational treatments of noncommuting symmetry flows and their associated 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.
Paper Structure (23 sections, 23 theorems, 245 equations)

This paper contains 23 sections, 23 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Background: Lagrangian structure of Liouville integrable systems
  3. Multi-time Euler-Lagrange equations
  4. Exterior derivative
  5. Higher forms
  6. A Lie group as multi-time
  7. Jet bundles over a Lie group
  8. Variational principle for functions on a Lie group
  9. Lagrangian 1-forms on Lie groups
  10. Building a 1-form from symmetries of a given Lagrangian
  11. Building a 1-form from Hamiltonians
  12. Nonlinear Poisson relations
  13. Exterior derivative and Poisson bracket
  14. Example: Kepler problem
  15. Example: Calogero-Moser system
  16. ...and 8 more sections

Key Result

Proposition 3.4

The field $z: G \rightarrow T^*Q$ is a symmetry group solution if and only if where $e$ is the unit element of the Lie group $G$.

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 3.1
  • Example 3.2
  • Example 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Definition 3.6
  • Definition 3.7
  • ...and 46 more