On the geometry of Lagrangian one-forms

TL;DR

This paper introduces a symplectic geometric framework for Lagrangian multiform theory, providing a unified approach to integrable systems that simplifies derivations and extends to non-commuting flows and Hamiltonian group actions.

Contribution

A new symplectic geometric formulation of Lagrangian multiform theory that treats all coordinates equally and extends to non-commuting flows and Hamiltonian actions.

Findings

Streamlined derivation of multi-time Euler-Lagrange equations.

Framework recasts any Lagrangian one-form in the new setting.

Extension to non-commuting flows and Hamiltonian Lie group actions.

Abstract

Lagrangian multiform theory is a variational framework for integrable systems. In this article we introduce a new formulation which is based on symplectic geometry and which treats position, momentum and time coordinates of a finite-dimensional integrable hierarchy on an equal footing. This formulation allows a streamlined one-step derivation of both the multi-time Euler-Lagrange equations and the closure relation (encoding integrability). We argue that any Lagrangian one-form for a finite-dimensional system can be recast in our new framework. This framework easily extends to non-commuting flows and we show that the equations characterising (infinitesimal) Hamiltonian Lie group actions are variational in character. We reinterpret these equations as a system of compatible non autonomous Hamiltonian equations.