On the geometry of Lagrangian one-forms

TL;DR

The paper addresses variational descriptions of finite-dimensional integrable hierarchies using Lagrangian one-forms. It develops a phase-space, symplectic, univariational principle with a Lagrangian one-form $L=p_ u dq^ u-H_i dt^i$ that places dependent and independent variables on equal footing. It proves the equivalence to the traditional two-step formulation via a Legendre transform, and extends the framework to nonabelian Hamiltonian group actions by leveraging Maurer–Cartan structure and moment maps, with explicit examples including the harmonic oscillator and Toda chain. The approach provides a unified geometrical perspective, enabling potential advances in quantization and extensions to integrable field theories.

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.